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# Maths A - Chapter 12

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### Maths A - Chapter 12

1. 1. 12syllabussyllabusrrefefererenceence Strand: Statistics and probability Core topic: Exploring and understanding data In thisIn this chachapterpter 12A Informal description of chance 12B Sample space 12C Tree diagrams 12D Equally likely outcomes 12E Using the fundamental counting principle 12F Relative frequency 12G Single event probability 12H Writing probabilities as decimals and percentages 12I Range of probabilities 12J Complementary events Introduction to probability MQ Maths A Yr 11 - 12 Page 487 Thursday, July 5, 2001 11:45 AM
2. 2. 488 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Introduction Sam’s dream is to win lotto and be instantly swept into the millionaire set. Each week he submits his six numbers to Q-lotto (an imaginary system) and anxiously awaits the results of the draw. Many Australians subscribe each week to the various systems, some using the same numbers, week after week, year after year, hoping one day that their numbers will ‘come up’; but what chance does each entry have of winning? How many numbers, and what combinations of numbers are necessary to win any prize? Will Sam win a million dollars, even if his six numbers do come up? Are there some numbers that appear to be chosen more frequently than others? Lotto systems are complex. We will endeavour to understand a simpliﬁed version of them. In a Q-lotto draw, balls numbered 1 to 45 are placed in a barrel and agitated in order to mix them thoroughly. Six balls are selected at random, in succession. It is these six numbers that entitle an entrant to ﬁrst prize (the jackpot prize), or a share in ﬁrst prize if there is more than one entry with these six correct numbers. Another two balls are then randomly selected. These two numbers are called bonus or supplementary numbers. Either of these two supplementary numbers may be used with three, four or ﬁve of the previous six numbers to give lower level prizes (called division prizes). Visit a lotto web site, which will display a table similar to the following one, which advises requirements for the various prize divisions. Combinations that win There are ﬁve prize divisions in Q-lotto. This table explains what you need to win. Any potential prize obviously depends on the jackpot size (the total amount of money available to be distributed to all ﬁrst-prize winners), and the number of winners in each of the divisions. So what is the chance that Sam will win money in any of the divisions? A web site will display a table similar to this, showing the chance of winning in any prize division. It appears that Sam’s chance of becoming a millionaire is quite remote (1 in 8.145 million, in fact)! Chances of winning Even if Sam happens to pick the correct six numbers, he will not necessarily win a million dollars. Examine the following Q-lotto draws for three consecutive weeks. Note Division Numbers required to win Division 1 Division 2 Division 3 Division 4 Division 5 All 6 winning numbers Any 5 winning numbers plus either supplementary Any 5 winning numbers Any 4 winning numbers Any 3 winning numbers plus either supplementary Division Chance based on four games Division 1 Division 2 Division 3 Division 4 Division 5 1 in 8.145 million 1 in 679 000 1 in 37 000 1 in 733 1 in 211 MQ Maths A Yr 11 - 12 Page 488 Thursday, July 5, 2001 10:21 AM
3. 3. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 489 that in Week 1, when the jackpot prize was over three million dollars, there were only three winners, so each received over one million dollars. In Week 2, the jackpot was twenty-four million dollars. There were 30 winners, so each winner’s share was less than one million dollars. The four winners in Week 3 received the same as each of the 30 winners in Week 2. To gain a better understanding of systems such as this, we need to understand the theory of chance and probability. After considering several simpler examples, we shall return to continue our investigation of Sam’s chance of fulﬁlling his dream. 1 Classify the chance of the following events occurring on a scale of impossible to certain, then position each on the scale at right. a Drawing a black card from a deck of cards b Winning the lotto c The sun rising tomorrow d A 6 will turn up on one roll of a 6-sided die e A Head will result if a coin is tossed f A total of 15 will result if two 6-sided dice are rolled. 2 Consider rolling a normal 6-sided die. a How many faces are on the die? b List the numbers on the die. c What would be the chance of rolling a 6? d How many even numbers are on the die? e What would be the chance of rolling an even number? f How many of the numbers are prime? g What would be the chance of rolling a number which was not prime? Data Draw 1 Draw 2 Draw 3 Date Week 1 Week 2 Week 3 Numbers 20, 30, 38, 40, 41, 43 5, 11, 22, 33, 40, 45 4, 5, 8, 15, 22, 30 Bonus nos. 1, 29 7, 12 23, 35 No. of winners 3 30 4 Jackpot size \$3 600 000 \$24 000 000 \$3 200 000 6 out of 6 paid: \$1 200 000 \$800 000 \$800 000 5 out of 6 + bonus nos. paid: \$14 000 \$10 000 \$12 000 5 out of 6 paid: \$1300 \$1000 \$1100 4 out of 6 paid: \$40 \$350 \$30 3 out of 6 + either bonus no. paid: \$25 \$20 \$20 Impossible Certain MQ Maths A Yr 11 - 12 Page 489 Thursday, July 5, 2001 10:21 AM
4. 4. 490 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 3 In a family there are 3 children. a Copy and complete the following table to show all possibilities of the order of birth of the children. b How many different outcomes are possible? c In how many cases are all 3 of the children of the same sex? d If there are both girls and boys in the family, how many situations occur where girls outnumber boys? 4 Convert the following fractions to decimals (to 3 decimal places, if necessary). a b c d e 5 Convert the fractions in question 4 to percentages (to 1 decimal place). 6 Calculate the following, giving your answer in both fraction and decimal form. a × × b ( )2 c 1 − d 1 − ( )2 e 1 − ( )3 f ( )2 × ( )3 Child 1 Child 2 Child 3 Boy Girl Girl Boy Boy Girl Terms used in chance Resources: Pen, paper, newspapers, other print material, other forms of media, World Wide Web or Internet, library. This activity would be best performed in groups. We frequently hear words such as: ‘There is absolutely no chance . . .’ ‘I am certain . . .’ ‘The chance of . . . is very slim.’ ‘. . . a ﬁfty-ﬁfty chance.’ These terms represent an expression of the chance that a particular event might occur. Task 1 1 As a group, make a list of as many terms of chance as you can. 2 Rank these terms in order of the likelihood of their occurring (some terms you may consider to be of equivalent chance of occurrence to each other). 3 Draw a line graph and place your terms in an appropriate position on this line. 4 Compare your graph with those compiled by other groups. 5 6 --- 2 3 --- 1 6 --- 4 52 ------ 5 12 ------ 1 2 --- 1 2 --- 1 2 --- 2 3 --- 1 8 --- 1 4 --- 1 2 --- 3 4 --- 1 4 --- inv estigat ioninv estigat ion MQ Maths A Yr 11 - 12 Page 490 Thursday, July 5, 2001 10:21 AM
5. 5. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 491 Informal description of chance You have booked a ski holiday to Thredbo for the middle of July. What is the chance that there will be enough snow on the ground for you to ski? There is no exact answer to this question, but by looking at the amount of snow in Thredbo during July over past years, we know that there is a very good chance that there will be enough snow to ski again this year. We can say that it is very likely that we will be able to ski during July at Thredbo. Terms such as very likely, almost certain, unlikely and ﬁfty-ﬁfty are used in everyday language to describe the chance of an event occurring. For the purposes of probability, an event is the outcome of an experiment that we are interested in. We can describe an outcome as a possible result to the probability experiment. Imagine you are playing a board game and it is your turn to roll the die. To win the game you need to roll a number less than 7. If you roll one die you must get a number less than 7. We would describe the chance of this event occurring as certain. When an event is certain to occur, the probability of that event occurring is 1. Now let’s consider an impossible situation. In a board game you have one last throw of the die. To win you must roll a 7. We know that this cannot be done. We would say that this is impossible. When an event is impossible, the probability of the event is 0. Task 2 The media (printed material, radio, television) often exaggerate events (positively or negatively) in their reports of incidents. 1 Search for and collect statements from a range of media where terms of chance have been used. Comment on the appropriateness of their use. Task 3 The English language has many colourful expressions to describe the chance of an event occurring. 1 Consider the following expressions, and research them through the Word Wide Web and your library, then answer the questions that follow. A ‘That will happen once in a blue moon.’ a What is a blue moon? b How often does a blue moon occur? B ‘There is Buckley’s chance of that happening.’ a Who was Buckley? b How did this saying originate? 2 Investigate to ﬁnd any similar expressions. What are their origins? MQ Maths A Yr 11 - 12 Page 491 Thursday, July 5, 2001 10:21 AM
6. 6. 492 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d The chance of any event occurring will often be somewhere between being certain and impossible and we use a variety of terms to describe where the chance lies in this range as shown in the ﬁgure at right. We use these terms based on our general knowledge of the world, the total possible outcomes and how often an event occurs. You will need to use terms of chance such as those used above to describe events that are more likely to occur than others. The term frequency refers to how often an event occurs. We use our knowledge about possible outcomes to order outcomes from the most frequent to the least frequent. Very unlikely Impossible Unlikely Fifty-fifty Probable Certain Almost certain 1– 2 1 0 SkillS HEET 12.1 Describe the chance of each of the following events occurring. a Tossing a coin and its landing Heads. b Rolling a 6 with one die. c Winning the lottery. d Selecting a numbered card from a standard deck of playing cards. THINK WRITE a There is an equal chance of the coin landing Heads and Tails. a The chance of tossing a Head is ﬁfty-ﬁfty. b There is only one chance in six of rolling a 6. b It is unlikely that you will roll a 6. c There is only a very small chance of winning the lottery. c It is very unlikely that you will win the lottery. d There are more numbered cards than picture cards in a deck. d It is probable that you will select a numbered card. 1WORKEDExample Mrs Graham is expecting her baby to be born between July 20 and 26. Is it more likely that her baby will be born on a weekday or a weekend? THINK WRITE There are 5 chances that the baby will be born on a weekday and 2 chances that it will be born on a weekend. It is more likely that Mrs Graham’s baby will be born on a weekday. 2WORKEDExample MQ Maths A Yr 11 - 12 Page 492 Thursday, July 5, 2001 10:21 AM
7. 7. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 493 In the above examples, we have been able to calculate which event is more likely by counting the number of ways an event may occur. This is not always possible. In some cases we need to use general knowledge to describe the chance of an event occurring. Consider the following probability problems. ‘The letters of the alphabet are written on cards and one card is selected at random. Which letter has the greatest chance of being chosen, E or Q?’ Each letter has an equal chance of being chosen because there is one chance that E will be chosen and one chance that Q will be chosen. ‘Stacey sticks a pin into a page of a book and she writes down the letter nearest to the pin. Which letter has the greater chance of being chosen, E or Q?’ This question is more difﬁcult to answer because each letter does not occur with equal frequency in written text. However, we know from our experience with the English language that Q will occur much less often than most other letters. We can therefore say that E has a greater chance than Q of being chosen. This is an example of using your knowledge of the world to make predictions about which event is more likely to occur. In this way, we make predictions about everyday things such as the weather and which football team will win on the weekend. A card is chosen from a standard deck. List the following outcomes in order from least likely to most likely: selecting a picture card, selecting an Ace, selecting a diamond, selecting a black card. THINK WRITE There are 12 picture cards in the deck. There are 4 Aces in the deck. There are 13 diamonds in the deck. There are 26 black cards in the deck. The order of events in ascending order of likeli- hood: selecting an Ace selecting a picture card selecting a diamond selecting a black card. 1 2 3 4 3WORKEDExample During the 1999 NRL season, the Sydney Roosters won 10 of their ﬁrst 12 games. In Round 13 they played the Northern Eagles who had won 3 of their ﬁrst 12 games. Who would be more likely to win? THINK WRITE Sydney Roosters have won more games than the Eagles. Sydney Roosters would be more likely to win, based on their previous results. (Footynote: The Eagles won the game 36–14. Sydney was more likely to win the game but nothing in football is certain.) 4WORKEDExample MQ Maths A Yr 11 - 12 Page 493 Thursday, July 5, 2001 10:21 AM
8. 8. 494 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d This is one example of past results being used to predict future happenings. There are many other such examples. Informal description of chance 1 Describe the chance of each of the following events occurring, using an appropriate probability term. a Selecting a ball with a double-digit number from a bag with balls numbered 1 to 40 b Selecting a female student from a class with 23 boys and 7 girls c Selecting a green marble from a barrel with 40 blue marbles and 30 red marbles d Choosing an odd number from the numbers 1 to 100 2 For each of the events below, describe the chance of it occurring as impossible, unlikely, even chance (ﬁfty-ﬁfty), probable or certain. a Rolling a die and getting a negative number b Rolling a die and getting a positive number c Rolling a die and getting an even number d Selecting a card from a standard deck and getting a red card e Selecting a card from a standard deck and getting a numbered card f Selecting a card from a standard deck and getting an Ace g Reaching into a moneybox and selecting a 30c piece h Selecting a blue marble from a bag containing 3 red, 3 green and 6 blue marbles Weather records show that it has rained on Christmas Day 12 times in the last 80 years. Describe the chance that it will rain on Christmas Day this year. THINK WRITE It has rained only 12 times on the last 80 Christmas Days. This is much less than half of all Christmas Days. It is unlikely that it will rain on Christmas Day this year. 5WORKEDExample remember 1. The chance of an event occurring ranges from being certain to impossible. 2. (a) An event that is certain has a probability of 1. (b) An event that is impossible has a probability of 0. 3. There are many terms that we use to describe the chance of an event occurring, such as improbable, unlikely, ﬁfty-ﬁfty, likely and probable. 4. Sometimes we can describe the chance of an event occurring by counting the possible outcomes, while other times we need to rely on our general knowledge to make such a description. remember 12A WORKED Example 1 MQ Maths A Yr 11 - 12 Page 494 Thursday, July 5, 2001 10:21 AM
9. 9. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 495 3 Give an example of an event which has a probability that could be described as: 4 Is it more likely that a person’s birthday will occur during a school term or during the school holidays? 5 For each event on the left, state whether it is more likely, less likely or equally likely to occur than the event on the right. a Fine weather Christmas Day Wet weather Christmas Day b A coin landing Heads A coin landing Tails c Rolling a total of 3 with two dice Rolling a total of 7 with two dice d Winning a rafﬂe made up of 50 tickets Winning a rafﬂe made up of 200 tickets e Winning a prize in the Lotto draw Not winning a prize in the Lotto draw 6 A die is thrown and the number rolled is noted. List the following events in order from least likely to most likely. Rolling an even number Rolling a number less than 3 Rolling a 6 Rolling a number greater than 2 7 Write the following events in order from least to most likely. Winning a rafﬂe with 5 tickets out of 30 Rolling a die and getting a number less than 3 Drawing a green marble from a bag containing 4 red, 5 green and 7 blue marbles Selecting a court card (Ace, King, Queen, Jack) from a standard deck Tossing a coin and having it land Heads 8 Before meeting in the cricket World Cup in 1999, Australia had beaten Zimbabwe in 10 of the last 11 matches. Who would be more likely to win on this occasion? 9 Which of the following two runners would be expected to win the ﬁnal of the 100 metres at the Olympic Games? Carl Bailey — best time 9.92 s and won his semi-ﬁnal Ben Christie — best time 10.06 s and 3rd in his semi-ﬁnal Give an explanation for your answer. 10 A stack of 26 cards has the letters of the alphabet written on them. Vesna draws a card from that stack. The probability of selecting a card that has a vowel written on it could best be described as: 11 Which of the following events is the most likely to occur? A Selecting the ﬁrst number drawn from a barrel containing 20 numbered marbles B Selecting a diamond from a standard deck of cards C Winning the lottery with one ticket out of 150 000 D Drawing the inside lane in the Olympic 100-metre ﬁnal with eight runners a certain b probable c even chance d unlikely e impossible. A unlikely B even chance C probable D almost certain WORKED Example 2 WORKED Example 3 WORKED Example 4 mmultiple choiceultiple choice mmultiple choiceultiple choice MQ Maths A Yr 11 - 12 Page 495 Thursday, July 5, 2001 10:21 AM
10. 10. 496 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 12 The ski season opens on the ﬁrst weekend of June. At a particular ski resort there has been sufﬁcient snow for skiing on that weekend on 32 of the last 40 years. Which of the following statements is true? A It is unlikely to snow at the opening of the ski season this year. B There is a ﬁfty-ﬁfty chance that it will snow at the opening of the ski season this year. C It is probable that it will snow at the opening of the ski season this year. D It is certain to snow at the opening of the ski season this year. 13 On a production line, light globes are tested to see how long they will last. After testing 1000 light globes it is found that 960 will burn for more than 1500 hours. Wendy purchases a light globe. Describe the chance of the light globe burning for more than 1500 hours. 14 Of 12 000 new cars sold last year, 1500 had a major mechanical problem during the ﬁrst year. Edwin purchased a new car. Describe the chance of Edwin’s car having a major mechanical problem in the ﬁrst year. 15 During an election campaign, 2000 people were asked for their voting preferences. One thousand said that they would vote for the government. If one person is chosen at random, describe the chance that he or she would vote for the government. Sample space At some time in our lives, most of us have tossed or will toss a coin. Many sports begin with the toss of a coin. What is the chance that the coin will land showing Heads? Most people would cor- rectly say ﬁfty-ﬁfty. We need to develop a method of accurately describing the prob- ability of an event. Before we can calculate probability, we need to be able to list all possible outcomes in a situation. This is called listing the sample space. When tossing a coin, the sample space has two elements: Heads and Tails. To calculate a probability, we need to know the number of elements in the sample space and what the elements are. mmultiple choiceultiple choice WORKED Example 5 MQ Maths A Yr 11 - 12 Page 496 Thursday, July 5, 2001 11:54 AM
11. 11. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 497 In many cases, several elements of the sample space may be the same. In such cases, we can distinguish between the number of elements in the sample space and the number of distinct (different) elements. In some situations there may be more than one element in the sample space that gives us the desired outcome. Favourable outcomes are the elements from the sample space that will meet the requirements for an event to occur. List the sample space for rolling a die. THINK WRITE The sample space is the numbers 1 to 6. S = {1, 2, 3, 4, 5, 6} 6WORKEDExample In a barrel there are 4 red marbles, 5 green marbles and 3 yellow marbles. One marble is drawn from the barrel. a List the sample space. b How many elements are in the sample space? c How many distinct elements are in the sample space? THINK WRITE a List each marble in the barrel. a S = {red, red, red, red, green, green, green, S = green, green, yellow, yellow, yellow} b Count the number of elements in the sample space. b The sample space has 12 elements. c Count the number of different elements in the sample space. c The sample space has 3 distinct elements. 7WORKEDExample Tegan is playing a board game. To win the game, Tegan must roll a number greater than 2 with one die. a List the sample space. b List the favourable outcomes. THINK WRITE a List all possible outcomes for one roll of a die. a S = {1, 2, 3, 4, 5, 6} b List all elements of the sample space which are greater than 2. b E = {3, 4, 5, 6} 8WORKEDExample MQ Maths A Yr 11 - 12 Page 497 Thursday, July 5, 2001 10:21 AM
12. 12. 498 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Sample space 1 The numbers 1 to 10 are written on cards that are turned face down. The cards are shufﬂed and one is chosen. List the sample space. 2 For each of the following probability experiments, state the sample space. a Tossing a coin b Rolling a die c The total when rolling two dice d Choosing a letter of the alphabet e The day of the week on which a baby could be born f The month in which a person’s birthday falls 3 For each of the following probability experiments, state the number of elements in the sample space. a Choosing a card from a standard deck b Selecting the winner of a 15-horse race c Selecting the ﬁrst ball drawn in a Lotto draw (The balls are numbered 1–45.) d Drawing a rafﬂe ticket from tickets numbered 1 to 1500 e Selecting a number in the range 100 to 1000, inclusive f Drawing a ball from a bag containing 3 yellow, 4 red and 4 blue balls 4 The letters of the word MISSISSIPPI are written on cards and turned face down. A card is then selected at random. a List the sample space. b How many elements has the sample space? c How many distinct elements in the sample space? 5 A card is to be selected from a standard deck. a How many elements has the sample space? b How many different elements has the sample space if we are interested in: i the suit of the card? ii the colour of the card? iii the face value of the card? remember 1. The sample space is the list of all possible outcomes in a probability experiment. 2. The number of elements in a sample space is the total number of possible outcomes. 3. In the sample space, there are sometimes several elements that are the same. We may be asked to count the number of distinct (different) elements in the sample space. 4. Favourable outcomes are the elements from the sample space which meet the requirements for a certain event to occur. remember 12B WORKED Example 6 WORKED Example 7 MQ Maths A Yr 11 - 12 Page 498 Thursday, July 5, 2001 10:21 AM
13. 13. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 499 6 Jane is playing a game of snakes and ladders. It is her turn to roll the die and to win she needs a number greater than 4. a List the sample space for this roll of the die. b List the favourable outcomes for this roll of the die. 7 A bag holds 60 black marbles and 40 white marbles. Tony is to choose one of these marbles from the bag. Tony wants to select a white marble. a How many elements are in the sample space? b How many favourable outcomes are contained in the sample space? 8 A bag contains 5 blue discs, 9 red discs and 6 yellow discs. To win a game, Jenny needs to draw a yellow disc from the bag. How many elements are in the sample space? 9 To win a game Jenny needs to draw a yellow disc from the bag in question 8. How many favourable outcomes are there? 10 A rafﬂe has 100 tickets. Chris buys 5 tickets in the rafﬂe. Which of the following statements is correct? A There are two elements in the sample space. B There are ﬁve favourable outcomes. C There are 100 elements in the sample space. D Both B and C 11 New South Wales are playing Queensland in a State of Origin match. a List the sample space for the possible outcomes of the match. b How many elements are in the sample space? c Is each element of the sample space equally likely to occur? Explain your answer. 12 A bag contains ﬁve 20c pieces, four 10c pieces and one 5c piece. A coin is selected at random from the bag. Without replacing the ﬁrst coin, a second coin is then selected. a List the sample space for the ﬁrst coin selected. How many elements has the sample space? b Assume that the ﬁrst coin chosen was a 20c piece. List the sample space for the second coin chosen. c Assume that the ﬁrst coin chosen was a 10c piece. List the sample space for the second coin chosen. d Assume that the ﬁrst coin chosen was the 5c piece. List the sample space for the second coin chosen. 13 Write down an example of an event that has 4 elements in the sample space. 14 Write down an example of an event that has 10 elements in the sample space but only 4 distinct elements. A 3 B 6 C 14 D 20 A 3 B 6 C 14 D 20 WORKED Example 8 mmultiple choiceultiple choice mmultiple choiceultiple choice mmultiple choiceultiple choice MQ Maths A Yr 11 - 12 Page 499 Thursday, July 5, 2001 10:21 AM
14. 14. 500 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 1 A die is rolled. Describe the chance that the uppermost face is 4. 2 A card is drawn from a standard pack. Describe the chance of selecting a black card. 3 A bag contains four \$1 coins and seven \$2 coins. Describe the chance that a coin drawn at random from the bag will be a \$2 coin. 4 A barrel containing balls numbered 1 to 100 has one ball selected at random from it. How many elements has the sample space? 5 Five history books, three reference books and ten sporting books are arranged on a shelf. A book is chosen at random from the shelf. How many elements are in the sample space? 6 For the example in question 5, how many distinct elements has the sample space? 7 For the example in question 5, if you want a sporting book, how many favourable outcomes are there? 8 Copy and complete. If an event is certain, then the probability of it occurring is . 9 Copy and complete. If an event is impossible, then the probability of it occurring is . 10 If Jane needs to select an Ace from a standard deck to win the game, how many favourable outcomes are there? 1 MQ Maths A Yr 11 - 12 Page 500 Thursday, July 5, 2001 11:51 AM
15. 15. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 501 Matching actual and expected results Resources: Coin, six-sided die, calculator. This investigation is best performed in small groups. Task 1 Consider tossing a coin once. 1 List the sample space. 2 Theoretically, if you tossed the coin 60 times, how many Heads and how many Tails would you expect to obtain? 3 In practice, these results may vary. Set up an experiment to determine the outcomes. Within your group, toss a coin 60 times and record the outcomes by copying and completing the table below: 4 How closely do your experimental results match the theoretical ones? 5 Combine your results with those of another group. These ﬁgures then represent the tossing of a coin 120 times. How do they compare with what you would expect in theory? 6 Collate and combine the results of all of the groups in your class. Compare these ﬁgures with what you would expect in theory. Have you reached any conclusions? Task 2 Repeat the processes in Task 1 using a six-sided die in place of the coin. Tabulate your results, comparing the theoretical and experimental outcomes. What conclusions can you draw? Task 3 Your calculator can be set to generate random integers. If you are unsure how to do this, your teacher will advise you. 1 Tossing a coin can be simulated on a calculator by the random generation of a 1 (representing, say a Tail) or a 2 (representing a Head). Using your calculator, repeat Task 1, simulating tossing a coin 60 times. 2 Set your calculator to generate the random integers 1, 2, 3, 4, 5 or 6. This can be used to simulate the rolling of a die. Repeat Task 2. inv estigat ioninv estigat ion Outcome Tally H T GCpr ogram Random GCpr ogram Dice 1 GCpr ogram Coin flip MQ Maths A Yr 11 - 12 Page 501 Thursday, July 5, 2001 10:21 AM
16. 16. 502 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Computer simulation:Tossing a coin and rolling a die Resources: Computer spreadsheet. Task 1 This task uses a computer to simulate tossing a coin 60 times. The following instructions refer to the Excel spreadsheet. If you use an alternative spreadsheet your teacher will advise you of the equivalent instructions and formulas. 1 In the spreadsheet, type the entries shown in cells A1, A3, D3, G3, D4, G4, D5 and G5. The ﬁgures in H4 and H5 are generated by the results of the experiment. Leave these two cells blank at this stage. 2 Enter the expected results of the experiment in cells E4 and E5. 3 Type in the heading shown in cell A7. 4 The 60 cells in the range A9 to J14 represent the results of the simulation of the 60 coin tosses. In these cells we are going to generate integers 1 or 2 randomly. If a 2 results, we will let that represent a Head and if a 1 results, we will say that represents a Tail. The formula to generate these Heads or Tails randomly is: =IF(INT(RAND()*2+1)=2,”Head”,”Tail”) Type this formula in cell A9 and copy it to all cells in the range A9 to J14. This will randomly generate a Head or a Tail in each of these cells every time the program is run. investigat ioninv estigat ionEXCE L Spreadshe et Random number generator EXCE L Spreadshe et Simulating coin tosses DIY EXCE L Spreadshe et Simulating coin tosses MQ Maths A Yr 11 - 12 Page 502 Thursday, July 5, 2001 10:21 AM
17. 17. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 503 5 The number of Heads and Tails generated are to be counted and the results displayed in cells H4 and H5. In cell H4 enter the formula =COUNTIF(A9:J14,”Head”) In cell H5 enter the formula =COUNTIF(A9:J14,”Tail”) 6 The function key F9 causes the computer to simulate the 60 tosses of the coin. Press the key and note the count of the resulting number of Heads and Tails in cells H4 and H5 respectively. Press the F9 key again. Record the results over a number of simulations. How do these ﬁgures compare with the theoretical expected results? Task 2 Repeat the previous experiment, simulating the rolling of a die 60 times. 1 Enter the expected numbers of 1s, 2s, etc in cells E4–E9. 2 To generate the integers 1, 2, 3, 4, 5 or 6 randomly in cells A13 to J18, a enter, in cell A13, the formula: =INT(RAND()*6+1) b Copy the formula to the other cells. 3 To count the numbers of 1s, 2s, etc in cell H4, enter the formula =COUNTIF(A13:J18,1) in cell H5, enter the formula =COUNTIF(A13:J18,2) . . . and so on. 4 Run the simulation a number of times (by pressing F9). Record and accumulate the results. Comment on the outcomes. Task 3 Formally record the results of this investigation in report form. EXCE L Spreadshe et Die MQ Maths A Yr 11 - 12 Page 503 Thursday, July 5, 2001 10:21 AM
18. 18. 504 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Tree diagrams A multi-stage event is an event where there is more than one part to the probability experiment. Tree diagrams are used to ﬁnd the elements in the sample space in a multi- stage probability experiment. Consider the case of tossing two coins. How many elements are there in the sample space? We draw a tree diagram to develop a system that will list the sample space for us. The tree diagram branches out once, for every stage of the probability experiment. At the end of each branch, one element of the sample space is found by following the branches that lead to that point. All of these elements together give us the outcomes of the experiment. Therefore, when two coins are tossed, the sample space can be written: S = {Head–Head, Head–Tail, Tail–Head, Tail–Tail} There are four elements in the sample space: Head–Tail and Tail–Head are distinct elements of the sample space. In many cases, the second branch of the tree diagram will be different from the ﬁrst branch. This occurs in situations such as those outlined in the following worked examples, where the ﬁrst event has an inﬂuence on the second event. The card chosen ﬁrst can then not be chosen in the second event. Head Head Tail Head Tail Tail 1st coin 2nd coin Head–Head Head–Tail Tail–Head Tail–Tail Outcomes A coin is tossed and a die is rolled. List all elements of the sample space. THINK WRITE Draw the branches for the coin toss. From each branch for the coin toss, draw the branches for the die roll. List the sample space by following the path to the end of each branch. S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, S = T5, T6} 1 2 Head Tail Coin toss Die roll 1 2 3 4 5 6 1 2 3 4 5 6 Outcomes H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6 3 9WORKEDExample MQ Maths A Yr 11 - 12 Page 504 Thursday, July 5, 2001 10:22 AM
19. 19. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 505 Each question must be read carefully, to see if repetition is possible or not. In the above example, the numbers cannot be repeated because we are drawing two cards without replacing the ﬁrst card. In examples such as tossing two coins, it is possible for the same outcome on both coins. When drawing a tree diagram, the tree needs to branch once for every stage of the experiment. When we roll two dice, there are two levels to the tree diagram. If we were to toss three coins, there would be three levels to the diagram, as shown at right. The numbers 2, 4, 7 and 8 are written on cards and are chosen to form a two-digit number. List the sample space. THINK WRITE Draw the ﬁrst branch of the tree diagram to show each possible ﬁrst digit. Draw the second branch of the tree diagram to show each possible second digit. When drawing the second branch, the digit from which the tree branches can’t be repeated. List the sample space by following the tree to the end of each branch. S = {24, 27, 28, 42, 47, 48, 72, 74, 78, 82, 84, 87} Note that in this case, 24 is not the same as 42. 1 2 4 7 8 1st digit 2nd digit 4 7 8 2 7 8 2 4 8 2 4 7 2 3 4 10WORKEDExample Head Head Head Tail Head Tail Tail Head Tail Tail Head Tail Head Tail 1st coin 2nd coin 3rd coin A coin showing ‘Heads’ MQ Maths A Yr 11 - 12 Page 505 Thursday, July 5, 2001 10:22 AM
20. 20. 506 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d For a family of four children: a Draw a tree diagram to list all possible combinations of boys and girls. b How many elements are in the sample space? c How many elements of the sample contain 3 boys and a girl? THINK WRITE a Draw the tree diagram. a b List the sample space by following the paths to the end of each branch. b S = {BBBB, BBBG, BBGB, BBGG, BGBB, BGBG, BGGB, BGGG, GBBB, GBBG, GBGB, GBGG, GGBB, GGBG, GGGB, GGGG} There are 16 elements in the sample space. c Count the number of elements that contain 3 boys and 1 girl. c There are four elements of the sample space which contain 3 boys and 1 girl. Boy Boy Boy Boy Girl Boy Girl Boy Girl Boy Girl Girl Girl Girl Boy Girl Boy Girl Boy Girl Boy Girl Boy Girl Boy Girl Boy Girl Boy Girl 2nd child1st child 3rd child 4th child 11WORKEDExample remember 1. A tree diagram is necessary in any example where there is more than one stage to the probability experiment. 2. The tree diagram must branch out once for every stage of the probability experiment. 3. Once the tree is drawn, the sample space is found by following the branches to each end. remember MQ Maths A Yr 11 - 12 Page 506 Thursday, July 5, 2001 10:22 AM
21. 21. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 507 Tree diagrams 1 Two coins are tossed. Use a tree diagram to list the sample space. 2 On three red cards, the numbers 1, 2 and 3 are written. On three blue cards, the same numbers are written. A red card and a blue card are then chosen to form a two-digit number. Draw a tree diagram to list the sample space. 3 A family consists of 3 children. Use a tree diagram to list all possible combinations of boys and girls. 4 A coin is tossed and then a die is rolled. a How many elements are in the sample space? b Does it make any difference to the sample space if the die is rolled ﬁrst and then the coin is tossed? 5 The digits 1, 3, 4 and 8 are written on cards. Two cards are then chosen to form a two- digit number. List the sample space. 6 Darren, Zeng, Melina, Kate and Susan are on a committee. From among themselves, they must select a chairman and a secretary. The same person cannot hold both positions. Use a tree diagram to list the sample space for the different ways the two positions can be ﬁlled. 7 A tennis team consists of six players, three males and three females. The three males are Andre, Pat and Yevgeny. The three females are Monica, Stefﬁ and Lindsay. A male and a female must be chosen for a mixed doubles match. Use a tree diagram to list the sample space. 8 Chris, Aminta, Rohin, Levi and Kiri are on a Landcare group. Two of them are to represent the group on a ﬁeld trip. Use a tree diagram to list all the different pairs that could be chosen. [Hint: In this example, a pairing of Chris and Aminta is the same as a pairing of Aminta and Chris.] 9 Four coins are tossed into the air. a Draw a tree diagram for this experiment. b Use your tree to list the sample space. c How many elements have an equal number of Heads and Tails? 10 Three coins are tossed into the air. The number of elements in the sample space is: 11 A two-digit number is formed using the digits 4, 6 and 9. If the same number can be repeated, the number of elements in the sample space is: 12 A two-digit number is formed using the digits 4, 6 and 9. If the same number cannot be used twice, the number of elements in the sample space is: A 3 B 6 C 8 D 9 A 3 B 6 C 8 D 9 A 3 B 6 C 8 D 9 12C WORKED Example 9 WORKED Example 10 WORKED Example 11 mmultiple choiceultiple choice mmultiple choiceultiple choice mmultiple choiceultiple choice MQ Maths A Yr 11 - 12 Page 507 Thursday, July 5, 2001 10:22 AM
22. 22. 508 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 13 When two coins are tossed there are three elements in the sample space, 2 Heads, 2 Tails or 1 Head and 1 Tail. Is this statement correct? Explain why or why not. 14 A two-digit number is to be formed using the digits 2, 5, 7 and 8. a If the same number can be used twice, list the sample space. b If the same number cannot be repeated, list the sample space. 15 The numbers 1, 2, 5 and 8 are written on cards and placed face down. a If two cards are chosen and used to form a two-digit number, how many elements are in the sample space? b If three cards are chosen and used to form a three-digit number, how many elements are in the sample space? c How many four-digit numbers can be formed using these digits? 16 A school captain and vice-captain need to be elected. There are ﬁve candidates. The three female candidates are Tracey, Jenny and Svetlana and the male candidates are Richard and Mushtaq. a Draw a tree diagram to ﬁnd all possible combinations of captain and vice-captain. b How many elements has the sample space? c If boys are ﬁlling both positions, how many elements are there? d If girls are ﬁlling both positions, how many elements are there? e If students of the opposite sex ﬁll the positions, how many elements are there? 17 Two dice are rolled. a Use a tree diagram to calculate the number of elements in the sample space. b Steve is interested in the number of elements for each total. Copy and complete the table below. c How many elements of the sample space have a two-digit number? Equally likely outcomes Below is the ﬁeld for the 1999 Melbourne Cup. Source: Courier-Mail 2 November 1999 Total 2 3 4 5 6 7 8 9 10 11 12 No. of elements MELBOURNE CUP ODDS Horse Latest odds Horse Latest odds 1 Tie The Knot 2 Central Park 3 Sky Heights 4 Maridpour 5 Arena 6 Yavana’s Pace 7 The Hind 8 Travelmate 9 Skybeau 10 The Message 11 Streak 12 Lahar 10-1 40-1 13-4 80-1 15-1 33-1 10-1 15-2 66-1 125-1 16-1 66-1 13 Second Coming 14 Able Master 15 Bohemiath 16 Figurehead 17 Rogan Josh 18 Laebeel 19 Brew 20 Lady Elsie 21 Rebbor 22 The Warrior 23 Zabuan 24 Zazabelle 66-1 30-1 25-1 100-1 5-1 7-1 33-1 60-1 50-1 50-1 125-1 80-1 MQ Maths A Yr 11 - 12 Page 508 Thursday, July 5, 2001 10:22 AM
23. 23. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 509 There are 24 horses in the race. The sample space therefore has 24 elements. How- ever, in this case, each outcome is not equally likely. This is because each horse in the race is not of equal ability. Some horses have a greater chance of winning than others. It is true in many practical situations that each outcome is not equally likely to occur. The weather on any day could be wet or ﬁne. Each outcome is not equally likely as there are many factors to consider, such as the time of year and the current weather patterns. In each probability example, it is important to consider whether or not each outcome is equally likely to occur. In general, when the selection is made randomly then equally likely outcomes will result. In some cases we need to use tree diagrams to calculate if each outcome is equally likely. A statement may seem logical, but unless further analysis is conducted, we cannot be sure. In a football match between Brisbane and Parramatta there are three possible outcomes: Brisbane win, Parramatta win and a draw. Is each outcome equally likely? Explain your answer. THINK WRITE Each team may not be of equal ability and draws occur less often than one of the teams winning. Each outcome is not equally likely as the teams may not be of equal ability and draws are fairly uncommon in football. 12WORKEDExample MQ Maths A Yr 11 - 12 Page 509 Thursday, July 5, 2001 10:22 AM
24. 24. 510 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Two-stage experiments (tossing two coins) Resources: Pen, paper, two coins, calculator, computer spreadsheet. This investigation would be best performed in pairs. Task 1 Consider the theoretical tossing of two coins. 1 Draw a tree diagram to show all possible outcomes. 2 List the elements of the sample space. Consider outcomes of Head–Tail and Tail–Head both to be equivalent to an outcome of 1 Head and 1 Tail. Is each outcome equally likely? 3 Consider tossing a pair of coins 36 times. In theory, how many of each outcome should result? Task 2 Consider an experiment consisting of tossing two coins 36 times. 1 Each pair of participants should have two coins. Draw up the following table to record the tossing of the pair of coins 36 times. When two coins are tossed there are three possible outcomes, 2 Heads, 2 Tails and one of each. Is each outcome equally likely? THINK WRITE There is more than one coin being tossed and so a tree diagram must be drawn. There are actually four outcomes, two of which involve 1 Head and 1 Tail. Therefore each of the outcomes mentioned is not equally likely to occur. Each outcome is not equally likely. There are two chances of getting one Head and one Tail. There is only one chance of getting 2 Heads and one chance of getting 2 Tails. 1 Head Head Tail Head Tail Tail 1st coin 2nd coin 2 13WORKEDExampleinv estigat ioninv estigat ion Outcome Tally HH HT/TH TT MQ Maths A Yr 11 - 12 Page 510 Thursday, July 5, 2001 10:22 AM
25. 25. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 511 2 Compare your tallies with the theoretical results you predicted in Task 1. 3 Compare your results with those obtained by other pairs in the class. Task 3 Set your calculator to generate random integers 1 or 2. Let a 1 represent a Tail and a 2 represent a Head. 1 With each person in the pair generating these random integers, simulate tossing two coins 36 times. Record your results in a table as you did in Task 2. 2 Compare these results with those obtained in Task 1 and Task 2. 3 Compare your answer with those from other pairs in the class. Task 4 This task uses a spreadsheet to simulate the tossing of the coins. Instructions and formulas relate to the Excel spreadsheet. If you are using a different one, your teacher will advise you of variations. 1 In the spreadsheet, type the entries shown in cells A1, E1, A3, D3, G3, E4, H4, E5, H5, E6, H6 and A8. Leave the cells I4, I5 and I6 blank at this stage. The results generated by the computer will be displayed there. 2 Enter the expected tally for each outcome in cells F4, F5 and F6. 3 The computer can randomly generate a 1 to represent a Tail or a 2 to represent a Head. To simulate the tossing of 2 coins, in the cell A10 enter the formula: =IF(INT(RAND()*2+1)+INT(RAND()*2+1)=4,”HH”,IF(INT(RAND()*2+1) +INT)RAND()*2+1)=2,”TT”,”HT/TH”)) Copy this formula to the 36 cells in the range A10:I13. MQ Maths A Yr 11 - 12 Page 511 Thursday, July 5, 2001 10:22 AM
26. 26. 512 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 4 To count the number of HH outcomes which result, in cell I4 enter the formula =COUNTIF(A10:I13,”HH”) For the HT/TH count, in the cell I5 enter the formula =COUNTIF(A10:I13,”HT/TH”) For the TT tally, in the cell I6, enter the formula =COUNTIF(A10:I13,”TT”) 5 Simulate the 36 tosses by pressing the F9 key. Note the results. Repeat the simulation until you have accumulated the tallies of 10 experiments. Comment on the results. Rolling two 6-sided dice Resources: Pen, paper, two 6-sided dice, calculator, computer spreadsheet. This activity is best performed in pairs and is designed as a follow-on from the investigation of tossing two coins. It relies on the skills developed previously, with only minimal guidance provided. (If you experience difﬁculties, consult the previous investigations where detailed information is supplied.) Approach this activity as you would an alternative assessment item, recording results in an orderly manner, culminating in the production of a report detailing your results. Task 1 Consider rolling two 6-sided dice 36 times. Explain the theoretical outcomes you would expect. Task 2 As a pair, roll two 6-sided dice 36 times. Record your results. Task 3 Use a calculator each to simulate the rolling of the two dice 36 times. Record your results. Task 4 Simulate the rolling of the two dice 36 times with the aid of a spreadsheet. Note your results. Reporting Write up the results of this investigation in the form of a report which must indicate detailed understanding of: a the theoretically expected results b the technique involved in undertaking the practical experiment and the results obtained c formulas used in the calculator simulation and the tallies resulting d formulas and results for the computer simulation — include at least two printouts showing the diversity of results e comparison of results from all tasks f summary and conclusion. inv estigat ioninv estigat ionGCpr ogram Dice 2 EXCE L Spreadshe et Dice MQ Maths A Yr 11 - 12 Page 512 Thursday, July 5, 2001 10:22 AM
27. 27. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 513 Equally likely outcomes 1 A tennis match is to be held between Lindsay and Anna. There are two possible out- comes, Lindsay to win and Anna to win. Is each outcome equally likely? Explain your answer. 2 There are 80 runners in the Olympic Games marathon. The sample space for the winner of the race therefore has 80 elements. Is each outcome equally likely? Explain your answer. 3 The numbers 1 to 40 are written on 40 marbles. The marbles are then placed in a bag and one is chosen from the bag. There are 40 elements to the sample space. Is each outcome equally likely? Explain your answer. 4 For each of the following, state whether each element of the sample space is equally likely to occur. a A card is chosen from a standard deck. b The result of a volleyball game between two teams. c It will either rain or be dry on a summer’s day. d A rafﬂe with 100 tickets has one ticket drawn to win ﬁrst prize. 5 For each of the following, state whether the statement made is true or false. Give a reason for your answer. a Twenty-six cards each have one letter of the alphabet written on them. One card is then chosen at random. Each letter of the alphabet has an equal chance of being selected. b A book is opened on any page and a pin is stuck in the page. The letter closest to the pin is then noted. Each letter of the alphabet has an equal chance of being selected. 6 In which of the following is each member of the sample space equally likely to occur? A Kylie’s softball team is playing a match that they could win, lose or draw. B A bag contains 4 red counters and 2 blue counters. One counter is selected from the bag. C The temperature on a January day will be between 20°C and 42°C. D A rose is planted in the garden that may bloom to be red, yellow or white. remember 1. Each element of the sample space will not always be equally likely. 2. Outcomes will be equally likely if a selection is random. When other factors inﬂuence the selection, each outcome is not equally likely. 3. When there is more than one event involved, examine the tree diagram to determine if events described are equally likely. remember 12D WORKED Example 12 mmultiple choiceultiple choice MQ Maths A Yr 11 - 12 Page 513 Thursday, July 5, 2001 10:22 AM
28. 28. 514 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 7 A couple have two children. They could have two boys, two girls or one of each. The sample space therefore has three elements that are all equally likely. Is this statement correct? Explain your answer. 8 In a game, two dice are rolled and the total of the two dice is the player’s score. a What is the sample space for the totals of two dice? b Is each element of the sample space equally likely to occur? 9 A restaurant offers a three course meal. The menu is shown below. a A diner selects one plate from each course. Draw a tree diagram to determine the number of elements in the sample space. b Is each element of the sample space equally likely to occur? 10 There are 10 horses in a race. Ken hopes to select the winner of the race. a How many elements in the sample space? b Is each element of the sample space equally likely to occur? Explain your answer. c Loretta selects her horse by drawing the names out of a hat. In this case, is the sample space the same? Is each element of the sample space equally likely to occur? Explain your answer. 1 Describe the chance of selecting an Ace from a standard deck of cards. For questions 2–5. A bag contains 3 black marbles, 4 white marbles and a red marble. 2 How many elements in the sample space? 3 How many distinct elements in the sample space? 4 If Julie needs to draw a red marble from the bag, how many favourable outcomes are there? 5 Is each element of the sample space equally likely to occur? 6 A pair of twins is born. Draw a tree diagram and then list the sample space for all possible combinations of boys and girls. 7 Amy and Samantha are in Year 11, while Luke, Matthew and John are in Year 12. One Year 11 student and one Year 12 student are to represent the school at a conference. List the sample space for all pairs that could be chosen. 8 A two-digit number is formed using the digits 1, 2, and 3. How many elements has the sample space if the same digit can be used twice? 9 A two-digit number is formed using the digits 5, 6, and 7. How many elements in the sample space if the same digit cannot be used twice? 10 A student has an exam in mathematics that she could either pass or fail. Is each element of the sample space equally likely to occur? WORKED Example 13 Entree Main course Prawn cocktail Oysters Soup Seafood platter Chicken supreme Roast beef Vegetarian quiche Dessert Pavlova Ice-cream 2 MQ Maths A Yr 11 - 12 Page 514 Thursday, July 5, 2001 10:22 AM
29. 29. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 515 Using the fundamental counting principle A three-course meal is to be served at a 21st birthday party. Guests choose one plate from each course, as shown in the menu below. In how many different ways can the three courses for the meal be chosen? There are two possible choices of entree, four choices for main course and three dessert choices. To ﬁnd the sample space for all possible outcomes, we draw a tree diagram. By following the path to the end of each branch we can see that there are 24 elements in the sample space. If we simply need to know the number of elements in the sample space, we multiply the number of possible choices at each level. Number of elements = 2 × 4 × 3 = 24 There are 24 ways in which the three-course meal can be chosen. This multiplication principle is called the fundamental counting principle. The total number of ways that a succession of choices can be made is found by multiplying the number of ways each single choice could be made. The fundamental counting principle is used when each choice is made independently of every other choice. That is, when one selection is made it has no bearing on the next selection. In the case above, the entree that is chosen has no bearing on what main course or dessert is chosen. Entree Main course Beef broth Calamari Spaghetti Roast chicken Pasta salad Grilled fish Dessert Ice-cream Banana split Strawberries Spaghetti Beef broth Roast chicken Pasta salad Grilled fish Main courseEntree Dessert Ice-cream Banana split Strawberries Ice-cream Banana split Strawberries Ice-cream Banana split Strawberries Ice-cream Banana split Strawberries Spaghetti Calamari Roast chicken Pasta salad Grilled fish Ice-cream Banana split Strawberries Ice-cream Banana split Strawberries Ice-cream Banana split Strawberries Ice-cream Banana split Strawberries MQ Maths A Yr 11 - 12 Page 515 Thursday, July 5, 2001 10:22 AM
30. 30. 516 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d A poker machine has three wheels. There are 20 symbols on each wheel. In how many different ways can the wheels of the poker machine ﬁnish, once they have been spun? THINK WRITE There are 20 possibilities for how the ﬁrst wheel can ﬁnish, 20 for the second wheel and 20 for the third wheel. Multiply each of these possibilities together. Total outcomes = 20 × 20 × 20 = 8000 Give a written answer. There are 8000 different ways in which the wheels of the poker machine can land. 1 2 14WORKEDExample In Year 11 at Blackhurst High School, there are four classes with 20, 22, 18 and 25 students in them respectively. A committee of four people is to be chosen, one from each class to represent Year 11 on the SRC. In how many ways can this group of four people be chosen? THINK WRITE There are 20 possible choices from the ﬁrst class, 22 from the second, 18 from the third and 25 from the fourth class. Multiply these possibilities together. Total possible outcomes = 20 × 22 × 18 × 25 = 198 000 Give a written answer. The committee of four people can be chosen in 198 000 different ways. 1 2 15WORKEDExample MQ Maths A Yr 11 - 12 Page 516 Thursday, July 5, 2001 10:22 AM
31. 31. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 517 Sometimes we need to reconsider examples that have some type of restriction placed on the possible selections. If number plates consist of 3 letters and 3 digits, how many are possible if the ﬁrst letter must be T, U or V, and the ﬁrst digit cannot be 0 or 1? THINK WRITE There are 3 possible ﬁrst letters. There are 26 possible second and third letters. There are 8 possible ﬁrst digits. There are 10 possible second and third digits. Multiply all these possibilities together. Total number plates = 3 × 26 × 26 × 8 × 10 × 10 = 1 622 400 Give a written answer. There are 1 622 400 possible number plates under this system. 1 2 3 4 5 6 16WORKEDExample remember 1. The fundamental counting technique allows us to calculate the number of different ways that separate events can occur. 2. This method can be used only when each selection is made independently of the others. To use this method, we multiply the number of ways that each selection can be made. remember MQ Maths A Yr 11 - 12 Page 517 Thursday, July 5, 2001 10:22 AM
32. 32. 518 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Using the fundamental counting principle 1 A poker machine has four reels, with 15 symbols on each wheel. If the wheels are spun, in how many ways can they ﬁnish? 2 Consider each of the following events. a A 10c coin and a 20c coin are tossed. In how many ways can they land? b A red die and blue die are cast. How many ways can the two dice land? c A coin is tossed and a die is rolled. How many possible outcomes are there? 3 A briefcase combination lock has a combination of three dials, each with 10 digits. How many possible combinations to the lock are there? 4 In the game of Yahtzee, ﬁve dice are rolled. In how many different ways can they land? 5 Some number plates have two letters followed by 4 numbers. How many of this style of plate are possible? 6 Personalised number plates that have six symbols can be any combination of letters or digits. How many of these are possible? 7 A restaurant menu offers a choice of four entrees, six main courses and three desserts. If one extra choice is offered in each of the three courses, how many more combi- nations of meal are possible? 8 There are 86 students in Year 11 at Narratime High School. Of these, 47 are boys and 39 are girls. One boy and one girl are to be chosen as school captains. In how many different ways can the boy and girl school captain be chosen? 9 A travel agency offers Queensland holiday packages ﬂying from Sydney with QANTAS and Ansett; travelling in First, Business and Economy class to Brisbane, the Gold Coast, the Great Barrier Reef and Cairns for periods of 7, 10 and 14 days. How many holiday packages does the traveller have to choose from? A 1 000 000 B 17 576 000 C 308 915 776 D 2 176 782 336 A 3 B 68 C 72 D 140 12E WORKED Example 14 mmultiple choiceultiple choice mmultiple choiceultiple choice WORKED Example 15 MQ Maths A Yr 11 - 12 Page 518 Thursday, July 5, 2001 10:22 AM
33. 33. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 519 10 A punter at the racetrack tries to pick the Daily Double. This requires her to pick the winner of race 6 and race 7. How many selections of two horses can she make if there are: a 8 horses in each race? b 12 horses in each race? c 14 horses in race 6 and 12 in race 7? d 16 horses in race 6 and seven in race 7? e 24 horses in race 6 and 16 horses in race 7? 11 A poker machine has ﬁve wheels and 20 symbols on each wheel. a In how many ways can the wheels of the poker machine ﬁnish when spun? b There are 4 Aces on the ﬁrst wheel, 5 on the second wheel, 2 on the third wheel, 6 on the fourth wheel and 1 on the ﬁfth. In how many ways can ﬁve Aces be spun on this machine? 12 Radio stations on the AM band have a call sign of a digit from 2 to 9, followed by two letters. a How many radio stations could there be under this system? b In Qld all stations begin with a 4. How many stations are possible in Qld? 13 At a shoe store a certain pair of shoes can be bought in black, brown or grey; lace up or buckle up and in six different sizes. How many different pairs of shoes are poss- ible? 14 Home telephone numbers in Australia have eight digits. a How many possible home telephone numbers are there? b If a telephone number can’t begin with either a 0 or 1, how many are possible? c Freecall 1800 numbers begin with 1800 and then six more digits. How many of these are possible? d A certain mobile network has numbers beginning with 015 or 018 followed by six digits. How many numbers can this network have? 15 Madako can’t remember his PIN number for his bank account. He knows that it has four digits, does not begin with nine, is an odd number and that all digits are greater than ﬁve. How many possible PIN numbers could he try? 16 Postcodes in Australia begin with either 2, 3, 4, 5, 6, 7 or 8 followed by three more digits. How many of these postcodes can there be? 17 Nadia goes to a restaurant that has a choice of 8 entrees, 15 main courses and 10 des- serts. a How many combinations of entree, main course and dessert are possible? b Nadia is allergic to garlic. When she examines the menu she ﬁnds that three entrees and four main courses are seasoned with garlic. How many possible choices can she make without choosing a garlic dish? 18 Bill is trying to remember Tom’s telephone number. It has eight digits and Bill can remember that it starts with 963 and ﬁnishes with either a 4 or a 6. How many poss- ible telephone numbers could Tom try? 19 A representative from each of six classes must be chosen to go on a committee. There are four classes of 28 students, a class of 25 students and a class of 20 students. How many committees are possible? A 70 B 1000 C 7000 D 10 000 WORKED EExample 16 mmultiple choiceultiple choice Work SHEET 12.1 MQ Maths A Yr 11 - 12 Page 519 Thursday, July 5, 2001 10:22 AM
34. 34. 520 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Q-lotto: Frequencies Now that we have some understanding of the theory of chance, let us continue to investigate Sam’s chance of winning Q-lotto. Each Q-lotto entry card has spaces to complete 12 games. Each game consists of 45 boxes. Six of these boxes are selected and marked with a cross. Consider the following questions: 1 Describe, in words, Sam’s chance of selecting one of the 6 correct numbers from the 45 available. 2 Describe Sam’s chance of selecting the correct 6 numbers from the 45 available. 3 Is he more or less likely to select the 6 correct than he is to choose 1 correct number? 4 Is it possible to select the same number more than once in each game? 5 List the sample space for each game. How many elements are in this sample space? 6 Considering the Q-lotto draw for Week 3 (see beginning of chapter), list the favourable outcomes. 7 How many elements are there in the set of favourable outcomes? Does this number change from week to week? 8 Of these elements, are they all required to win a prize? If not, how many are required? 9 Which ones are required to win the Division 1 prize? 10 Does each of the 45 numbers have an equal chance of being selected? Why/ why not? Let us consider the following statistics of frequencies of draws of the 45 balls in Q-lotto. (B1 = Ball no. 1; B2 = Ball no. 2 etc.; shaded rows show the number of times the ball number has been drawn.) investigat ioninv estigat ion 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 Game 1 Number of times drawn B1 B2 B3 B4 B5 B6 B7 B8 B9 60 54 54 65 48 43 60 55 70 B10 B11 B12 B13 B14 B15 B16 B17 B18 56 70 51 60 49 62 59 64 64 B19 B20 B21 B22 B23 B24 B25 B26 B27 55 62 60 52 52 59 50 58 71 B28 B29 B30 B31 B32 B33 B34 B35 B36 76 71 62 59 52 49 65 69 57 B37 B38 B39 B40 B41 B42 B43 B44 B45 56 70 65 60 55 59 70 50 55 MQ Maths A Yr 11 - 12 Page 520 Thursday, July 5, 2001 10:22 AM
35. 35. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 521 Consider the following: 11 Do these tables reﬂect the numbers chosen by entrants? 12 How many times has the number 1 been drawn as one of the lucky numbers? 13 How many weeks is it since the number 1 was drawn? 14 What number has been drawn most frequently? How many times has it been drawn? 15 What number has been drawn least frequently? How many times has it been drawn? 16 Which numbers were drawn last week? You should have obtained eight numbers. Why are there 8 numbers and not only 6? 17 If you were to put in a Q-lotto entry next week, basing your numbers on those which have been chosen more than others, which numbers might you include in your 6? 18 Basing your Q-lotto entry next week on the fact that those numbers which haven’t turned up for a while might turn up next week, which numbers might you choose? We’re well on the way to answering the lotto questions posed at the beginning of this chapter. We’ll resume our investigation after we consider relative frequency. At that stage, we should be able to verify Sam’s chance of picking the 6 winning numbers as quoted earlier in the chapter. Number of weeks since drawn B1 B2 B3 B4 B5 B6 B7 B8 B9 3 13 3 5 13 4 6 10 3 B10 B11 B12 B13 B14 B15 B16 B17 B18 0 4 18 1 1 13 8 1 6 B19 B20 B21 B22 B23 B24 B25 B26 B27 1 0 0 3 1 9 0 6 1 B28 B29 B30 B31 B32 B33 B34 B35 B36 1 8 9 1 6 8 2 2 5 B37 B38 B39 B40 B41 B42 B43 B44 B45 10 0 2 0 0 3 0 6 6 MQ Maths A Yr 11 - 12 Page 521 Thursday, July 5, 2001 10:47 AM
36. 36. 522 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Relative frequency You are planning to go skiing on the ﬁrst weekend in July. The trip is costing you a lot of money and you don’t want your money wasted on a weekend without snow. So what is the chance of it snowing on that weekend? We can use past records only to estimate that chance. If we know that it has snowed on the ﬁrst weekend of July for 54 of the last 60 years, we could say that the chance of snow this year is very high. To measure that chance, we calculate the relative frequency of snow on that weekend. We do this by dividing the number of times it has snowed by the number of years we have examined. In this case, we can say the relative frequency of snow on the ﬁrst weekend in July is 54 ÷ 60 = 0.9. The relative frequency is usually expressed as a decimal and is calculated using the formula: Relative frequency = In this formula, a trial is the number of times the probability experiment has been conducted. The relative frequency is used to assess the quality of products. This is done by ﬁnding the relative frequency of defective products. number of times an event has occurred number of trials --------------------------------------------------------------------------------------------- The weather has been ﬁne on Christmas Day in Sydney for 32 of the past 40 Christmas Days. Calculate the relative frequency of ﬁne weather on Christmas Day. THINK WRITE Write the formula. Relative frequency = Substitute the number of ﬁne Christmas Days (32) and the number of trials (40). Relative frequency = Calculate the relative frequency as a decimal. = 0.8 1 number of times an event has occurred number of trials --------------------------------------------------------------------------------------------- 2 32 40 ------ 3 17WORKEDExample A tyre company tests its tyres and ﬁnds that 144 out of a batch of 150 tyres will withstand 20 000 km of normal wear. Find the relative frequency of tyres that will last 20 000 km. Give the answer as a percentage. THINK WRITE Write the formula. Relative frequency = Substitute 144 (the number of times the event occurred) and 150 (number of trials). Relative frequency = Calculate the relative frequency. = 0.96 Convert the relative frequency to a percentage. = 96% 1 number of times an event has occurred number of trials --------------------------------------------------------------------------------------------- 2 144 150 --------- 3 4 18WORKEDExample MQ Maths A Yr 11 - 12 Page 522 Thursday, July 5, 2001 10:47 AM
37. 37. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 523 Relative frequencies can be used to solve many practical problems. A batch of 200 light globes was tested. The batch is considered unsatisfactory if more than 15% of globes burn for less than 1000 hours. The results of the test are in the table below. Determine if the batch is unsatisfactory. No. of hours No. of globes Less than 500 4 500 to less than 750 12 750 to less than 1000 15 1000 to less than 1250 102 1250 to less than 1500 32 1500 or more 35 THINK WRITE Count the number of light globes that burn for less than 1000 hours. 31 light globes burn for less than 1000 hours. Write the formula. Relative frequency = Substitute 31 (number of times the event occurs) and 200 (number of trials). Relative frequency = Calculate the relative frequency. = 0.155 Convert the relative frequency to a percentage. = 15.5% Make a conclusion about the quality of the batch of light globes. More than 15% of the light globes burn for less than 1000 hours and so the batch is unsatisfactory. 1 2 number of times an event has occurred number of trials --------------------------------------------------------------------------------------------- 3 31 200 --------- 4 5 6 19WORKEDExample remember 1. The relative frequency is used to estimate the probability of an event. 2. The relative frequency, usually expressed as a decimal, is a ﬁgure which represents how often an event has occurred. 3. The relative frequency is calculated using the formula: Relative frequency = . 4. The relative frequency can also be written as a percentage and is used to solve practical problems. number of times an event has occurred number of trials --------------------------------------------------------------------------------------------- remember MQ Maths A Yr 11 - 12 Page 523 Thursday, July 5, 2001 10:47 AM
38. 38. 524 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Relative frequency 1 At the opening of the ski season, there has been sufﬁcient snow for skiing for 37 out of the past 50 years. Calculate the relative frequency of sufﬁcient snow at the begin- ning of the ski season. 2 A biased coin has been tossed 100 times with the result of 79 Heads. Calculate the relative frequency of the coin landing Heads. 3 Of eight maths tests done by a class during a year, Peter has topped the class three times. Calculate the relative frequency of Peter topping the class. 4 Farmer Jones has planted a wheat crop. For the wheat crop to be successful farmer Jones needs 500 mm of rain to fall over the spring months. Past weather records show that this has occurred on 27 of the past 60 years. Find the relative frequency of: a sufﬁcient rainfall b insufﬁcient rainfall. 5 Of 300 cars coming off an assembly line, 12 are found to have defective brakes. Calculate the relative frequency of a car having defective brakes. Give the answer as a percentage. 6 A survey of 25 000 new car buyers found that 750 cars had a major mechanical problem in the ﬁrst year of operation. Calculate the relative frequency of the car: a having mechanical problems in the ﬁrst year b not having mechanical problems in the ﬁrst year. 7 On a production line, light globes are tested to see how long they will last. After testing 1000 light globes, it is found that 960 will burn for more than 1500 hours. Wendy purchases a light globe. What is the relative frequency that the light globe will: a burn for more than 1500 hours? b burn for no more than 1500 hours? 8 A study of cricket players found that of 150 players, 36 batted left handed. What is the relative frequency of left-handed batsmen? 9 Four surveys were conducted and the following results were obtained. Which result has the highest relative frequency? A Of 1500 P-plate drivers, 75 had been involved in an accident. B Of 1200 patients examined by a doctor, 48 had to be hospitalised. C Of 20 000 people at a football match, 950 were attending their ﬁrst match. D Of 50 trucks inspected, 2 were found to be unroadworthy. A 0.24 B 0.36 C 0.64 D 0.76 12F WORKED Example 17 WORKED Example 18 mmultiple choiceultiple choice mmultiple choiceultiple choice MQ Maths A Yr 11 - 12 Page 524 Thursday, July 5, 2001 10:47 AM
39. 39. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 525 10 During an election campaign 2000 people were asked for their voting preferences. One thousand and ﬁfty said that they would vote for the government, 875 said they would vote for the opposition and the remainder were undecided. What is the relative frequency of: a government voters? b opposition voters? c undecided voters? 11 Research over the past 25 years shows that each November there is an average of two wet days on Sunnybank Island. Travelaround Tours offer one-day tours to Sunnybank Island at a cost of \$150 each, with a money back guarantee against rain. a What is the relative frequency of wet November days as a percentage? b If Travelaround Tours take 1200 bookings for tours in November, how many refunds would they expect to give? 12 An average of 200 robberies takes place each year in the town of Amiak. There are 10 000 homes in this town. a What is the relative frequency of robberies in Amiak? b Each robbery results in an average insurance claim of \$20 000. What would be the minimum premium that the insurance company would need to charge to cover these claims? 13 A car maker recorded the ﬁrst time that its cars came in for mechanical repairs. The results are in the table below. The assembly line will need to be upgraded if the relative frequency of cars needing mechanical repair in the ﬁrst year is greater than 25%. Determine if this will be neces- sary. 14 For the table in question 13 determine, as a percentage, the relative frequency of: a a car needing mechanical repair in the ﬁrst 3 months b a car needing mechanical repair in the ﬁrst 2 years c a car not needing mechanical repair in the ﬁrst 3 years. Time taken No. of cars 0 to <3 months 5 3 to <6 months 12 6 to <12 months 37 1 to <2 years 49 2 to <3 years 62 3 or more years 35 WORKED Example 19 MQ Maths A Yr 11 - 12 Page 525 Thursday, July 5, 2001 10:47 AM
40. 40. 526 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 15 A manufacturer of shock absorbers measures the distance that its shock absorbers can travel before they must be replaced. The results are in the table below. The relative frequency of the shock absorber lasting is 0.985 for a certain guaranteed distance. What is the maximum distance the manufacturer will guarantee? 16 A soccer team plays 40 matches over a season and the results (wins, losses and draws) are shown below. W W W D L L L D W L W D L D W W L L L D W W D L L W W W L D L D D L W W W D D L a Put this information into a table showing the number of wins, losses and draws. b Calculate the relative frequency of each result over a season. No. of kilometres No. of shock absorbers 0 to <20 000 1 20 000 to <40 000 2 40 000 to <60 000 46 60 000 to <80 000 61 80 000 to <100 000 90 Relative frequencies Resources: Calculator, World Wide Web. Task 1 1 Set your calculator to generate random integers in the range 1 to 10 inclusive. 2 Draw up the table below to record the result of generating 100 random numbers: 3 Calculate the relative frequency of each number. Is it what you would have expected? 4 Repeat the experiment once more. Comment on any differences. inv estigat ioninv estigat ion Integer Tally Frequency Relative frequency 1 2 3 4 5 6 7 8 9 10 MQ Maths A Yr 11 - 12 Page 526 Thursday, July 5, 2001 10:47 AM
41. 41. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 527 Single event probability Previously we discussed the chances of certain events occurring. In doing so, we used informal terms such as probable and unlikely. However, while these terms give us an idea of whether something is likely to occur or not, they do not tell us how likely they are. To do this, we need an accurate way of stating the probability. We stated earlier that the chance of any event occurring was somewhere between impossible and certain. We also said that: 1. if an event is impossible the probability was 0 2, if an event is certain the probability was 1. It therefore follows that the probability of any event must lie between 0 and 1 inclusive. A probability is a number that describes the chance of an event occurring. All prob- abilities are calculated as fractions but can be written as fractions, decimals or percent- ages. Probability is calculated using the formula: The total number of favourable outcomes is the number of different ways the event can occur, while the total number of outcomes is the number of elements in the sample space. Task 2 The Bureau of Meteorology has a web site detailing the temperature, rainfall, cloudy days etc. for a large number of towns throughout Queensland. This is found at: http://www.bom.gov.au/climate 5 Visit the web site and search to locate the town closest to where you live. Choose some aspect of the climate and compile a relative frequency table displaying the occurrence and variation of the weather in your area. Write a conclusion highlighting your ﬁndings. Is it consistent with your experiences? Task 3 The World Wide Web records an abundance of statistical information — for example, results of sporting teams, stock market movements, Melbourne Cup winners, world leaders, movie attendance. 6 Choose a topic which interests you and research its statistics through the Web. Draw up a relative frequency table and summarise its results. P event( ) number of favourable outcomes total number of outcomes ----------------------------------------------------------------------------= Zoran is rolling a die. To win a game, he must roll a number greater than 2. List the sample space and state the number of favourable outcomes. THINK WRITE There are 6 possible outcomes. S = {1, 2, 3, 4, 5, 6} The favourable outcomes are to roll a 3, 4, 5 or 6. There are 4 favourable outcomes. 1 2 20WORKEDExample MQ Maths A Yr 11 - 12 Page 527 Thursday, July 5, 2001 10:47 AM
42. 42. 528 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Consider the case of tossing a coin. If we are calculating the probability that it will land Heads, there is 1 favourable outcome out of a total of 2 possible outcomes. Hence we can then write P(Heads) = . This method is used to calculate the probability of any single event. In the above example the fraction could be simpliﬁed to . Some questions have more than one favourable outcome. In these cases, we need to add together each of these outcomes to calculate the number of outcomes that are favourable. 1 2 --- Andrea selects a card from a standard deck. Find the probability that she selects an Ace. THINK WRITE There are 52 cards in the deck (total number of outcomes). There are 4 Aces (number of favourable outcomes). Write the probability. P(Ace) = 1 2 3 4 52 ------ 21WORKEDExample 4 52 ------ 1 13 ------ In a barrel there are 6 red marbles, 2 green marbles and 4 yellow marbles. One marble is drawn at random from the barrel. Calculate the probability that the marble drawn is red. THINK WRITE There are 12 marbles in the barrel (total number of outcomes). There are 6 red marbles in the barrel (number of favourable outcomes). Write the probability. P(red) = = 1 2 3 6 12 ------ 1 2 --- 22WORKEDExample On a bookshelf there are 4 history books, 7 novels, 2 dictionaries and 5 sporting books. If I select one at random, what is the probability that the one chosen is not a novel? THINK WRITE There are 20 books on the shelf (total number of outcomes). Seven of these books are novels, meaning that 13 of them are not novels (number of favourable outcomes). Write the probability. P(not a novel) = 1 2 3 13 20 ------ 23WORKEDExample MQ Maths A Yr 11 - 12 Page 528 Thursday, July 5, 2001 10:47 AM
43. 43. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 529 Some questions do not require us to calculate the entire sample space, only the sample space for a small part of the experiment. The digits 1, 3, 4, 5 are written on cards and these cards are then used to form a four-digit number. Calculate the probability that the number formed is: a even b greater than 3000. THINK WRITE a If the number is even the last digit must be even. a There are four cards that could go in the ﬁnal place (total number of outcomes). Only one of these cards (the 4) is even (number of favourable outcomes). Write the probability. P(even) = b If the number is greater than 3000, then the ﬁrst digit must be a 3 or greater. b There are four cards that could go in the ﬁrst place. Three of these cards are a 3 or greater. Write the probability. P(greater than 3000) = 1 2 3 4 1 4 --- 1 2 3 4 3 4 --- 24WORKEDExample remember 1. The sample space is the list of all possible outcomes in a probability experiment. 2. The probability of an event is calculated using the formula: P(event) = number of favourable outcomes total number of outcomes ---------------------------------------------------------------------------- remember MQ Maths A Yr 11 - 12 Page 529 Thursday, July 5, 2001 10:47 AM
44. 44. 530 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Single event probability 1 A coin is tossed at the start of a cricket match. Manuel calls Heads. List the sample space and the number of favourable outcomes. 2 For each of the following probability experiments, list the sample space and state the number of favourable outcomes. a Rolling a die and needing a 6 b Rolling two dice and needing a total greater than 9 c Choosing a letter of the alphabet and it being a vowel d The chance a baby will be born on the weekend e The chance that a person’s birthday will fall in summer 3 For each of the following probability experiments, state the number of favourable outcomes and the total number of outcomes. a Choosing a red card from a standard deck b Selecting the winner of a 15-horse race c Selecting the ﬁrst ball drawn in a lotto draw (The balls are numbered 1 to 45.) d Winning a rafﬂe with 5 tickets out of 1500 e Selecting a yellow ball from a bag containing 3 yellow, 4 red and 4 blue balls 4 A coin is tossed. Find the probability that the coin will show Tails. 5 A regular die is rolled. Calculate the probability that the uppermost face is: 6 A barrel contains marbles with the numbers 1 to 45 written on them. One marble is drawn at random from the bag. Find the probability that the marble drawn is: a 6 b 1 c an even number d a prime number e less than 5 f at least 5. a 23 b 7 c an even number d an odd number e a multiple of 5 f a multiple of 3 g a number less than 20 h a number greater than 35 i a square number. 12G WORKED Example 20 WORKED Example 21 MQ Maths A Yr 11 - 12 Page 530 Thursday, July 5, 2001 10:47 AM
45. 45. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 531 7 Many probability questions are asked about decks of cards. You should know the cards making up a standard deck. A card is chosen from a standard deck. Find the probability that the card chosen is: 8 A bag contains 12 counters: 7 are orange, 4 are red and 1 is yellow. One counter is selected at random from the bag. Find the probability that the counter chosen is: 9 The digits 2, 3, 5 and 9 are written on cards. One card is then chosen at random. Find the probability that the card chosen is: 10 In a bag of fruit there are 4 apples, 6 oranges and 2 pears. Larry chooses a piece of fruit from the bag at random but he does not like pears. Find the probability that Larry does not select a pear. 11 The digits 2, 3, 5 and 9 are written on cards. They are then used to form a four-digit number. Find the probability that the number formed is: 12 A die is rolled. The probability that the number on the uppermost face is less than 4 is: a the Ace of diamonds b a King c a club d red e a picture card f a court card. a yellow b red c orange. a the number 2 b the number 5 c even d odd e divisible by 3 f a prime number. a even b odd c divisible by 5 d less than 3000 e greater than 5000. A B C D K Q J 10 9 8 7 6 5 4 3 2 2A K Q J 10 9 8 7 6 5 4 3 2 A 2 K Q J 10 9 8 7 6 5 4 3 2 A 2 K Q J 10 9 8 7 6 5 4 3 2 A 2 WORKED Example 22 WORKED Example 23 WORKED Example 24 mmultiple choiceultiple choice 1 6 --- 1 3 --- 1 2 --- 2 3 --- MQ Maths A Yr 11 - 12 Page 531 Thursday, July 5, 2001 12:09 PM
46. 46. 532 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 13 When a die is rolled, which of the following outcomes does not have a probability equal to ? A The number on the uppermost face is greater than 3. B The number on the uppermost face is even. C The number on the uppermost face is at least a 3. D The number on the uppermost face is a prime number. 14 A card is chosen from a standard deck. The probability that the card chosen is a court card is: 15 When a card is chosen from a standard deck, which of the following events is most likely to occur? 16 One thousand tickets are sold in a rafﬂe. Craig buys ﬁve tickets. a One ticket is drawn at random. The holder of that ticket wins ﬁrst prize. Find the probability of Craig winning ﬁrst prize. b After the ﬁrst prize has been drawn, a second prize is drawn. If Craig won ﬁrst prize, what is the probability that he now also wins second prize? 17 A lottery has 160 000 tickets. Janice buys one ticket. There are 3384 cash prizes in the lottery. a What is the probability of Janice winning a cash prize? b If there are 6768 consolation prizes of a free ticket for being one number off a cash prize, what is the probability that Janice wins a consolation prize? c What is the probability that Janice wins either a cash prize or a consolation prize? 18 A number is formed using all ﬁve of the digits 1, 3, 5, 7 and 8. What is the probability that the number formed: 19 Write down an example of an event which has a probability of: 20 A three-digit number is formed using the digits 2, 4 and 7. a Explain why it is more likely that an even number will be formed than an odd number. b Which is more likely to be formed: a number less than 400 or a number greater than 400? A B C D A choosing a seven B choosing a club C choosing a picture card D choosing a black card a begins with the digit 3 b is even c is odd d is divisible by 5 e is greater than 30 000 f is less than 20 000. a b c . mmultiple choiceultiple choice 1 2 --- mmultiple choiceultiple choice 1 52 ------ 1 13 ------ 3 13 ------ 4 13 ------ mmultiple choiceultiple choice 1 2 --- 1 4 --- 2 5 --- MQ Maths A Yr 11 - 12 Page 532 Thursday, July 5, 2001 10:47 AM
47. 47. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 533 1 A die is rolled. Find the probability that the uppermost face is 4. 2 A card is drawn from a standard pack. Find the probability of selecting a Jack. 3 A bag contains four \$1 coins and seven \$2 coins. Find the probability that a coin drawn at random from the bag will be a \$2 coin. 4 A barrel, containing balls numbered 1 to 100, has one ball selected at random from it. Find the probability that the ball selected is a multiple of 3. 5 Five history books, 3 reference books and 10 sporting books are arranged on a shelf. What is the probability of a sporting book being on the left-hand end of the bookshelf? 6 A coin is tossed 10 times with a result of 7 Heads and 3 Tails. The relative frequency of this coin landing Heads is 0.7; true or false? 7 A coin is tossed 10 times with a result of 7 Heads and 3 Tails. The probability of this coin landing Tails is ; true or false? 8 In 60 rolls of a die, there have been 12 sixes. What is the relative frequency of rolling a six? 9 During a football season a team has won 15 matches and lost 5. Calculate the relative frequency of the team winning. 10 A car assembly line ﬁnds that ﬁve in every 1000 cars have faulty paintwork. If I purchase one of these cars ﬁnd, as a percentage, the relative frequency that the paintwork is faulty. 3 7 10 ------ MQ Maths A Yr 11 - 12 Page 533 Thursday, July 5, 2001 12:09 PM
48. 48. 534 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Comparing probabilities with actual results In this activity, we compare the probability of certain events to practical results. You may be able to do a simulation of these activities on a spreadsheet. Tossing a coin If we toss a coin, P(Heads) = . 1 If you toss a coin, how many Heads would you expect in: a 4 tosses b 10 tosses c 50 tosses d 100 tosses. 2 Toss a coin 100 times and record the number of Heads after: a 4 tosses b 10 tosses c 50 tosses d 100 tosses. 3 Combine your results with those of the rest of the class. How close to 50% is the total number of Heads thrown by the class? Rolling a die 1 When you roll a die, what is the probability of rolling a 1? (In fact, the probability for each number on the die is the same.) 2 Roll a die 120 times and record each result in the table below. How close are the results to the results that were expected? Rolling two dice 1 Roll two dice and record the total on the faces of the two dice. Repeat this 100 times and complete the table below. 2 Do you notice anything different about the results of this activity, compared to the others? investigat ioninv estigat ion 1 2 --- Number Occurrences Percentage of throws 1 2 3 4 5 6 Number Occurrences Percentage of throws 2 3 4 5 6 7 8 9 10 11 12 MQ Maths A Yr 11 - 12 Page 534 Thursday, July 5, 2001 10:47 AM
49. 49. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 535 Writing probabilities as decimals and percentages In our exercises so far, we have been writing probabilities as fractions. This is the way that most mathematicians like to express chance. However, in day-to-day language, decimals and percentages are also used. Therefore, we need to be able to write prob- abilities as both decimals and percentages. When writing a probability as a decimal, we use the same formula and divide the numerator by the denominator to convert to a decimal. The chance of an event occurring is commonly expressed as a percentage. This is the percentage chance of an event occurring. When writing a probability as a percentage, we take the fractional answer and multiply by 100% to convert to a percentage. If I select a card from a standard deck, what is the probability of selecting a heart, expressed as a decimal? THINK WRITE There are a total of 52 cards in the deck (elements of the sample space). There are 13 hearts in the deck (elements of the event space). Write the probability. P(heart) = Convert to a decimal. = 0.25 1 2 3 13 52 ------ 4 25WORKEDExample In a bag there are 20 counters: 7 are green, 4 are blue and the rest are yellow. If I select one at random, ﬁnd the probability (as a percentage) that the counter is yellow. THINK WRITE There are 20 counters in the bag (elements of the sample space). There are 9 yellow counters in the bag (elements of the event space). Write the probability. P(yellow counter) = × 100% Convert to a percentage. = 45% 1 2 3 9 20 ------ 4 26WORKEDExample remember 1. Sometimes it is necessary to write a probability as a decimal or a percentage. 2. To write a probability as a decimal, we calculate the probability as a fraction, then divide the numerator by the denominator to convert to a decimal. 3. To write a probability as a percentage, we calculate the probability as a fraction, then multiply by 100% to convert to a percentage. remember MQ Maths A Yr 11 - 12 Page 535 Thursday, July 5, 2001 10:47 AM
50. 50. 536 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Writing probabilities as decimals and percentages 1 A die is rolled. What is the probability of rolling an even number, expressed as a decimal? 2 A barrel contains 40 marbles. There are 10 blue marbles, 15 red marbles and 15 white marbles. A marble is selected at random from the barrel. Calculate as a percentage the probability of selecting a red marble. 3 Write down the probability that a tossed coin will land Tails: a as a decimal b as a percentage. 4 A student is rolling a die. Write down each of the following probabilities as decimals, correct to 2 decimal places. a Getting a 1 b Getting an odd number c Getting a number greater than 4 5 For rolling a die, write down the following probabilities as percentages. Give your answers correct to 1 decimal place. a Getting a 3 b Getting an even number c Getting a number less than 6 6 From a standard deck of cards, one is selected at random. Write down the probability of each of the following as a decimal (correct to 2 decimal places where necessary). a Selecting the King of hearts b Selecting a spade c Selecting any 5 d Selecting a red card e Selecting a court card (any King, Queen or Jack — the Jack is also called a Knave) 7 When selecting a card from a standard deck, what would be the probabilities of the following, written as percentages? Give your answers correct to 1 decimal place. a Selecting a Jack of clubs b Selecting a diamond c Selecting any 2 d Selecting a black card e Selecting a court card 12H SkillS HEET 12.2 WORKED Example 25 SkillS HEET 12.3 WORKED Example 26 MQ Maths A Yr 11 - 12 Page 536 Thursday, July 5, 2001 10:47 AM
51. 51. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 537 8 A rafﬂe has 400 tickets. Sonya has bought 8 tickets. The probability that Sonya wins ﬁrst prize in the rafﬂe is: 9 In a class of 25 students, there are 15 boys and 10 girls. If a student is chosen at random from the class, the probability that the student is a boy is: 10 Which of the following does not describe the chance of selecting a diamond from a standard deck of cards? 11 The diagram on the right shows a spinner that can be used in a board game. When the player spins the spinner, what is the probability of getting the following results (expressed as a decimal)? a A 5 b An even number c An odd number d A number greater than one 12 The board game in question 11 has the following rules. A player spinnning a 2 or a 5 is out of the game. A player spinning a 3 collects a treasure and automatically wins the game. Write down the probability, as a percentage, that with the next spin a player: a wins the game b is out of the game c neither wins nor is out of the game. A 0.02 B 0.08 C 0.2 D 0.8 A 10% B 15% C 40% D 60% A B 0.13 C 0.25 D 25% mmultiple choiceultiple choice mmultiple choiceultiple choice mmultiple choiceultiple choice 13 52 ------ 1 2 3 4 5 MQ Maths A Yr 11 - 12 Page 537 Thursday, July 5, 2001 10:48 AM
52. 52. 538 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Range of probabilities Consider the following problem: A die is rolled. Calculate the probability that the uppermost face is a number less than 7. We know this is certain to occur but we will look at the solution using the probability formula. There are 6 elements in the sample space and 6 elements that are favourable. Therefore: P(no. less than 7) = = 1 When the probability of an event is 1, the event is certain to occur. Now let’s consider an impossible situation: A die is rolled. Calculate the probability that the uppermost face is a number greater than 7. There are 6 elements in the sample space and there are 0 elements that are favour- able. Therefore: P(no. greater than 7) = = 0 When the probability of an event is 0, the event is impossible. All probabilities therefore lie in the range 0 to 1. An event with a probability of has an even chance of occurring or not occurring. The range of probabilities can be seen in the ﬁgure at right. This ﬁgure allows us to make a connection between the formal probabilities that we calculated in the previous exercise, and the informal terms we used earlier in the chapter. The closer a probability is to 0, the less likely it is to occur. The closer the probability is to 1, the more likely it is to occur. 0 ≤ P(E) ≤ 1 We read this as: ‘The probability of an event is greater than or equal to zero, and less than or equal to 1’. 6 6 --- 0 6 --- Very unlikely Impossible Unlikely Fifty-fifty Probable Certain Almost certain 1– 2 1 0 1 2 --- For the following probabilities, describe whether the event would be certain, probable, ﬁfty-ﬁfty, unlikely or impossible. a b 0 c THINK WRITE a is less than and is therefore unlikely to occur. a The event is unlikely as it has a probability of less than . b A probability of 0 means the event is impossible. b The event is impossible as it has a probability of 0. c = . Therefore, the event has an even chance of occurring. c The event has an even chance of occurring as the probability = . 4 9 --- 18 36 ------ 4 9 --- 1 2 --- 1 2 --- 18 36 ------ 1 2 --- 1 2 --- 27WORKEDExample MQ Maths A Yr 11 - 12 Page 538 Thursday, July 5, 2001 10:48 AM
53. 53. C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 539 You need to be able to recognise when you can and cannot measure the probability. You cannot measure probability when each outcome is not equally likely. In a batch of 400 televisions, 20 are defective. If one television is chosen, ﬁnd the probability of its not being defective and describe this chance in words. THINK WRITE There are 400 televisions (elements of the sample space). There are 380 televisions which are not defective (number of favourable outcomes). Write the probability. P(not defective) = = Since the probability is much greater than and very close to 1, it is very probable that it will not be defective. It is very probable that the television chosen will not be defective. 1 2 3 380 400 --------- 19 20 ------ 4 1 2 --- 28WORKEDExample State whether the following statements are true or false, and give a reason for your answer. a The probability of correctly selecting a number between 1 and 10 drawn out of a barrel is . b The weather tomorrow could be ﬁne or rainy, therefore the probability of rain is . THINK WRITE a Each outcome is equally likely. a True, because each number is equally likely to be selected. b Each outcome is not equally likely. b False, because there is not an equal chance of the weather being ﬁne or rainy. 1 10 ------ 1 2 --- 29WORKEDExample remember 1. Probabilities range from 0 to 1. A probability of 0 means that the event is impossible, while a probability of 1 means the event is certain. 2. By calculating the probability, we are able to make a connection with the more informal descriptions of chance. 3. The rules of probability can be applied only when each outcome is equally likely to occur. remember MQ Maths A Yr 11 - 12 Page 539 Thursday, July 5, 2001 10:48 AM