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- 1. Chapter 10 Probability
- 2. Section 10.1 Simple Experiments <ul><li>Goals </li></ul><ul><ul><li>Study probability </li></ul></ul><ul><ul><ul><li>Experimental probability </li></ul></ul></ul><ul><ul><ul><li>Theoretical probability </li></ul></ul></ul><ul><ul><li>Study probability properties </li></ul></ul><ul><ul><ul><li>Mutually exclusive events </li></ul></ul></ul><ul><ul><ul><li>Unions and intersections of events </li></ul></ul></ul><ul><ul><ul><li>Complements of events </li></ul></ul></ul>
- 3. Interpreting Probability <ul><li>Probability is the mathematics of chance. </li></ul><ul><li>For example, the statement “The chances of winning the lottery game are 1 in 150,000” means that only 1 of every 150,000 lottery tickets printed is a winning ticket. </li></ul>
- 4. Probability Terminology <ul><li>Making an observation or taking a measurement is called an experiment . </li></ul><ul><li>An outcome is one of the possible results of an experiment. </li></ul><ul><li>The set of all possible outcomes is called the sample space . </li></ul><ul><li>An event is any collection of possible outcomes. </li></ul>
- 5. Example 1 <ul><li>An experiment consists of rolling a standard six-sided die and recording the number of dots showing on the top face. </li></ul><ul><ul><li>List the sample space. </li></ul></ul><ul><ul><li>List one possible event. </li></ul></ul>
- 6. Example 1 <ul><li>The sample space contains 6 possible outcomes: {1, 2, 3, 4, 5, 6}. </li></ul><ul><li>One possible event: The event of rolling an even number: {2,4,6} </li></ul>
- 7. Example 2 <ul><li>An experiment consists of tossing a coin 3 times and recording the results in order. </li></ul><ul><ul><li>List the sample space. </li></ul></ul><ul><ul><li>List one possible event. </li></ul></ul>
- 8. Example 2, cont’d <ul><li>Solution: The sample space contains 8 possible outcomes and can be written {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. </li></ul><ul><li>One possible event is {HTH, HTT, TTH, TTT}, which is the event of getting a tail on the second coin toss. </li></ul>
- 9. Example 3 <ul><li>An experiment consists of spinning a spinner twice and recording the colors it lands on. </li></ul><ul><ul><li>List the sample space. </li></ul></ul><ul><ul><li>List one possible event. </li></ul></ul>
- 10. Example 3 <ul><li>Solution: The sample space contains 16 possible outcomes and can be written {RR, RY, RG, RB, YR, YY, YG, YB, GR, GY, GG, GB, BR, BY, BG, BB}. </li></ul><ul><li>One possible event is {RR, YY, GG, BB}, which is the event of getting the same color on both spins. </li></ul>
- 11. Example 4 <ul><li>The experiment consists of rolling 2 standard dice and recording the number appearing on each die. </li></ul><ul><ul><li>List the sample space. </li></ul></ul><ul><ul><li>List one possible event. </li></ul></ul>
- 12. Example 4 <ul><li>Solution: The sample space contains 36 possible outcomes and can be written {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}. </li></ul>
- 13. Example 4 <ul><li>Solution, cont’d: One possible event is {(6,1), (5,2), (4,3), (3,4), (2,5), (1,6)}, which is the event of getting a total of 7 dots on the two dice. </li></ul>
- 14. Probability, cont’d <ul><li>The probability of an event is a number from 0 to 1, and can be written as a fraction, decimal, or percent. </li></ul><ul><ul><li>The greater the probability, the more likely the event is to occur. </li></ul></ul><ul><ul><li>An impossible event has probability 0. </li></ul></ul><ul><ul><li>A certain event has probability 1. </li></ul></ul>
- 15. Experimental Probability <ul><li>One way to find the probability of an event is to conduct a series of experiments. </li></ul>
- 16. Example 5 <ul><li>An experiment consisted of tossing 2 coins 500 times and recording the results </li></ul><ul><li>Let E be the event of getting a head on the first coin Find the experimental probability of E . </li></ul>
- 17. Example 5 <ul><li>The even of getting a head on the first coin: </li></ul><ul><li>{HH, HT}. </li></ul><ul><li>occurred a total of </li></ul><ul><li>137 + 115 = 252 times out of 500. </li></ul><ul><ul><li>The experimental probability of E is </li></ul></ul>
- 18. Theoretical Probability <ul><li>Another way to find the probability of an event is to use the theory of what “should” happen rather than conducting experiments. </li></ul>
- 19. Example 6 <ul><li>An experiment consists of tossing 2 fair coins. </li></ul><ul><li>Find the theoretical probability of: </li></ul><ul><ul><li>Each outcome in the sample space. </li></ul></ul><ul><ul><li>The event E of getting a head on the first coin. </li></ul></ul><ul><ul><li>The event of getting at least one head. </li></ul></ul>
- 20. Example 6, cont’d <ul><li>Solution: </li></ul><ul><ul><li>There are 4 outcomes in the sample space: {HH, HT, TH, TT}. Each outcome is equally likely to occur. </li></ul></ul>
- 21. Example 6 <ul><li>The event E is {HH, HT} and the theoretical probability of E is the number of outcomes in E divided by the number of outcomes in the sample space. </li></ul>
- 22. Example 6 <ul><li>c) The event of getting at least one head is E = {HH, HT, TH}. </li></ul>
- 23. Example 7 <ul><li>An experiment consists of rolling 2 fair dice. </li></ul><ul><li>Find the theoretical probability of: </li></ul><ul><ul><li>Event A : getting 7 dots. </li></ul></ul><ul><ul><li>Event B : getting 8 dots. </li></ul></ul><ul><ul><li>Event C : getting at least 4 dots. </li></ul></ul>
- 24. Example 7 <ul><li>Solution: There are 36 outcomes in the sample space. (see pg 635 for a list of the sample space.) </li></ul><ul><ul><li>Event A : getting 7 dots contains 6 outcomes: </li></ul></ul><ul><ul><li>{(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}. </li></ul></ul><ul><ul><li>Each outcome is equally likely to occur. </li></ul></ul>
- 25. Example 7 <ul><li>Solution, cont’d: </li></ul><ul><ul><li>Event B : getting 8 dots contains 5 equally likely outcomes: </li></ul></ul><ul><ul><li>{(2,6), (3,5), (4,4), (5,3), (6,2)}. </li></ul></ul>
- 26. Example 7 <ul><li>Solution, cont’d: </li></ul><ul><ul><li>Event C : getting at least 4 dots (or a total of 4 or more) contains 33 equally likely outcomes: </li></ul></ul><ul><ul><li>{ (1,3), (1,4), (1,5), (1,6), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) }. </li></ul></ul>
- 27. Example 8 <ul><li>A jar contains four marbles: </li></ul><ul><li>1 red, 1 green, 1 yellow, and 1 white. </li></ul>
- 28. Example 8 <ul><li>If we draw 2 marbles in a row, without replacing the first one, find the probability of: </li></ul><ul><ul><li>Event A : One of the marbles is red. </li></ul></ul><ul><ul><li>Event B : The first marble is red or yellow. </li></ul></ul><ul><ul><li>Event C : The marbles are the same color. </li></ul></ul><ul><ul><li>Event D : The first marble is not white. </li></ul></ul><ul><ul><li>Event E : Neither marble is blue. </li></ul></ul>
- 29. Example 8 <ul><li>If we draw 2 marbles in a row, without replacing the first one, find the probability of: </li></ul><ul><li>Event A : One of the marbles is red. </li></ul><ul><ul><li>A = {RG, RY, RW, GR, YR, WR}. </li></ul></ul>
- 30. Example 8 <ul><li>If we draw 2 marbles in a row, without replacing the first one, find the probability of: </li></ul><ul><ul><li>Event B : The first marble is red or yellow. </li></ul></ul><ul><ul><ul><li>B = {RG, RY, RW, YR, YG, YW}. </li></ul></ul></ul><ul><ul><li>Event C : The marbles are the same color. </li></ul></ul>
- 31. Example 8 <ul><li>If we draw 2 marbles in a row, without replacing the first one, find the probability of: </li></ul><ul><ul><li>Event D : The first marble is not white. </li></ul></ul><ul><ul><li>D = {RG, RY, RW, GR, GY, GW, YR, YG, YW}. </li></ul></ul><ul><ul><ul><li>e) Event E : Neither marble is blue. </li></ul></ul></ul>
- 32. Union and Intersection <ul><li>The union of two events, A U B , refers to all outcomes that are in one, the other, or both events. </li></ul><ul><li>The intersection of two events, A ∩ B, refers to outcomes that are in both events. </li></ul>
- 33. Mutually Exclusive Events <ul><li>Events that have no outcomes in common are said to be mutually exclusive . </li></ul><ul><li>If A and B are mutually exclusive events, then </li></ul>
- 34. Example 9 <ul><li>A card is drawn from a standard deck of cards. </li></ul><ul><ul><li>Let A be the event the card is a face card. </li></ul></ul><ul><ul><li>Let B be the event the card is a black 5. </li></ul></ul><ul><li>Find and interpret P( A U B ). </li></ul>
- 35. Example 9 <ul><li>Solution: The sample space contains 52 equally likely outcomes. (this chart is on pg 639) </li></ul>
- 36. Example 9 <ul><li>Event A has 12 outcomes, one for each of the 3 face cards in each of the 4 suits. </li></ul><ul><ul><li>P( A ) = 12/52. </li></ul></ul><ul><li>Event B has 2 outcomes, because there are 2 black fives. </li></ul><ul><ul><li>P( B ) = 2/52. </li></ul></ul><ul><li>Events A and B are mutually exclusive because it is impossible for a 5 to be a face card. </li></ul><ul><ul><li>P( A U B ) = 12/52 + 2/52 = 14/52 = 7/26. </li></ul></ul><ul><ul><li>This is the probability of drawing either a face card or a black 5. </li></ul></ul>
- 37. Complement of an Event <ul><li>The set of outcomes in a sample space S , but not in an event E , is called the complement of the event E . </li></ul><ul><ul><li>The complement of E is written Ē . </li></ul></ul><ul><ul><li>For example: The weather man says there is a 45% chance of rain. The complement of this event is the chance that it will not rain which is 55%. </li></ul></ul>
- 38. Complement of an Event, cont’d <ul><li>The relationship between the probability of an event E and the probability of its complement Ē is given by: </li></ul>
- 39. Example 10 <ul><li>In a number matching game, </li></ul><ul><ul><li>First Carolan chooses a whole number from 1 to 4. </li></ul></ul><ul><ul><li>Then Mary guesses a number from 1 to 4. </li></ul></ul><ul><li>What is the probability the numbers are equal? </li></ul><ul><li>What is the probability the numbers are unequal? </li></ul>
- 40. Example 10 <ul><li>Solution: The sample space contains 16 outcomes: { (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) }. </li></ul><ul><li>Let E be the event the numbers are equal. </li></ul><ul><ul><li>P( E ) = ¼ </li></ul></ul><ul><li>Then Ē (complement of E) is the event the numbers are unequal. </li></ul><ul><ul><li>P( Ē ) = 1 – ¼ = ¾ </li></ul></ul>
- 41. Example 12 <ul><li>An experiment consists of spinning the spinner once and recording the number on which it lands. </li></ul>
- 42. Example 12 <ul><li>Define 4 events: </li></ul><ul><ul><li>A: an even number </li></ul></ul><ul><ul><li>B: a number greater than 5 </li></ul></ul><ul><ul><li>C: a number less than 3 </li></ul></ul><ul><ul><li>D: a number other than 2 </li></ul></ul><ul><li>Find P( A ), P( B ), P( C ), and P( D ). </li></ul><ul><li>Find and interpret P( A U B ) and P( A ∩ B ). </li></ul><ul><li>Find and interpret P( B U C ) and P( B ∩ C ). </li></ul>
- 43. Example 12 <ul><li>Solution: The sample space has 8 equally likely outcomes: {1, 2, 3, 4, 5, 6, 7, 8}. </li></ul><ul><li>Find P( A ),P( B ), P( C ), and P( D ). </li></ul><ul><ul><li>A = {2, 4, 6, 8}, so P( A ) = 4/8 = 1/2 </li></ul></ul><ul><ul><li>B = {6, 7, 8}, so P( B ) = 3/8 </li></ul></ul><ul><ul><li>C = {1, 2}, so P( C ) = 2/8 = 1/4 </li></ul></ul><ul><ul><li>D = {1, 3, 4, 5, 6, 7, 8}, so P( D ) = 7/8. </li></ul></ul>
- 44. Example12 <ul><li>Solution, cont’d: </li></ul><ul><li>Find and interpret P( A U B ) and </li></ul><ul><li>P( A ∩ B ). </li></ul><ul><ul><li>A and B are not mutually exclusive, so </li></ul></ul><ul><ul><li>A ∩ B = {6, 8}, so P( A ∩ B ) = 2/8 </li></ul></ul>
- 45. Example 12 <ul><li>Solution, cont’d: </li></ul><ul><li>Find and interpret P( B U C ) and </li></ul><ul><li>P( B ∩ C ). </li></ul><ul><ul><li>B and C are mutually exclusive, so </li></ul></ul><ul><ul><li> and </li></ul></ul>
- 46. Section 10.2 Multistage Experiments <ul><li>Goals </li></ul><ul><ul><li>Study tree diagrams </li></ul></ul><ul><ul><ul><li>Study the fundamental counting principle </li></ul></ul></ul><ul><ul><li>Study probability tree diagrams </li></ul></ul><ul><ul><ul><li>Study the additive property </li></ul></ul></ul><ul><ul><ul><li>Study the multiplicative property </li></ul></ul></ul>
- 47. 10.2 Initial Problem <ul><li>A friend who likes to gamble wagers that if you toss a coin repeatedly you will get 2 tails before you get 3 heads. </li></ul><ul><li>Should you take the bet? </li></ul><ul><ul><li>The solution will be given at the end of the section. </li></ul></ul>
- 48. Example 1 <ul><li>A tree diagram is drawn for the experiment of drawing 1 ball from a box containing 1 red ball, 1 white ball, and 1 blue ball. </li></ul>
- 49. Example 2 <ul><li>There are 12 possible outcomes. </li></ul><ul><li>12 is the product of 4 branches times 3 branches. </li></ul>
- 50. Fundamental Counting Principle <ul><li>The number of outcomes in an experiment can also be determined using the fundamental counting principle : </li></ul><ul><ul><li>If an event or action A can occur in r ways, and , for each of these r ways, an event or action B can occur in s ways, then the number of ways events or actions A and B can occur, in succession, is r times s . </li></ul></ul><ul><ul><li>The principle can be extended to more than two events or actions. </li></ul></ul>
- 51. Example 3 <ul><li>The options on a pizza are: </li></ul><ul><ul><li>Small, medium, or large </li></ul></ul><ul><ul><li>White or wheat crust </li></ul></ul><ul><ul><li>Sausage, pepperoni, bacon, onion, mushrooms </li></ul></ul><ul><li>How many different one-topping pizzas are possible? </li></ul>
- 52. Example 3 <ul><li>Solution: </li></ul><ul><ul><li>The first action is choosing 1 of 3 sizes. </li></ul></ul><ul><ul><li>The second action is choosing 1 of 2 crusts. </li></ul></ul><ul><ul><li>The third action is choosing 1 of 5 toppings. </li></ul></ul><ul><li>There are 3(2)(5) = 30 different one-topping pizzas possible. </li></ul>
- 53. Example 4 <ul><li>Find the probability of getting a sum of 11 when tossing a pair of fair dice. </li></ul>
- 54. Example 4 <ul><li>Solution: There are 36 equally likely outcomes in the sample space. </li></ul><ul><ul><li>6 possible outcomes on the first roll </li></ul></ul><ul><ul><li>6 possible outcomes on the second roll </li></ul></ul><ul><ul><li>6(6) = 36 </li></ul></ul><ul><li>There are 2 ways of rolling a sum of 11: (5,6) and (6,5). </li></ul><ul><ul><li>The probability is 2/36 = 1/18. </li></ul></ul>
- 55. Probability Tree Diagrams <ul><li>Tree diagrams can also be used to determine probabilities in multistage experiments. </li></ul><ul><li>Tree diagrams that are labeled with the probabilities of events are called probability tree diagrams . </li></ul>
- 56. Probability Tree Diagrams
- 57. Example 6 <ul><li>Draw a probability tree diagram to represent the experiment of drawing one ball from a container holding 2 red balls and 3 white balls. </li></ul>
- 58. Example 6, cont’d <ul><li>Solution: The first tree has one branch for each ball. </li></ul><ul><li>The second tree was simplified by combining branches. </li></ul>
- 59. Example 7 <ul><li>Draw a probability tree diagram to represent the experiment of spinning the spinner once. </li></ul><ul><li>Find the probability of landing on white or on green. </li></ul>
- 60. Example 7 <ul><li>Solution: There are 4 outcomes in the sample space. </li></ul><ul><ul><li>They are not all equally likely. </li></ul></ul><ul><ul><li>They are all mutually exclusive. (no overlapping) </li></ul></ul><ul><ul><li>Use the central angles to find the probability of each outcome and draw the probability tree diagram. </li></ul></ul>
- 61. Example 7 <ul><li>The probability of white or green is: </li></ul>
- 62. Example 8 <ul><li>A jar contains 3 marbles, 2 black and 1 red. </li></ul><ul><li>A marble is draw and replaced, and then a second marble is drawn. What is the probability both marbles are black? </li></ul>
- 63. Example 8 <ul><li>Solution: Draw a probability tree diagram to represent the experiment. </li></ul>
- 64. Example 8 <ul><li>Assign a probability to the end of each secondary branch. (multiply fractions as you move down branches, add factions as you move across the end of branches.) </li></ul>
- 65. Example 8 <ul><li>Either tree diagram can be used to find that P( BB ) = 4/9. </li></ul>
- 66. Example 9 <ul><li>A jar contains 3 red balls and 2 green balls. </li></ul><ul><li>First a coin is tossed. </li></ul><ul><ul><li>If the coin lands heads, a red ball is added to the jar. </li></ul></ul><ul><ul><li>If the coin lands tails, a green ball is added to the jar. </li></ul></ul><ul><li>Second a ball is selected from the jar. </li></ul><ul><li>What is the probability a red ball is chosen? </li></ul>
- 67. Example 9 <ul><li>Solution: A probability tree diagram is created. </li></ul>
- 68. Example 9 <ul><li>The probability of choosing a red ball is found by adding the probabilities at the end of the branches labeled R . </li></ul>
- 69. Example 10 <ul><li>A jar contains 3 marbles, 2 black and 1 red. </li></ul><ul><li>A marble is drawn and not replaced before a second marble is drawn. </li></ul><ul><li>What is the probability that both marbles were black? </li></ul>
- 70. Example 10 <ul><li>Solution: Create a probability tree diagram to represent the experiment. </li></ul>
- 71. Example 10 <ul><li>The probability of choosing 2 black marbles is: </li></ul>
- 72. Example 11 <ul><li>A jar contains 2 red gumballs and 2 green gumballs. </li></ul><ul><li>An experiment consists of drawing gumballs one at a time from a jar until a red one is chosen. </li></ul><ul><li>Find the probability of: </li></ul><ul><ul><li>A: only 1 draw is needed </li></ul></ul><ul><ul><li>B: exactly 2 draws are needed </li></ul></ul><ul><ul><li>C: exactly 3 draws are needed </li></ul></ul>
- 73. Example 11 <ul><li>Solution: Create a probability tree diagram. </li></ul><ul><li>The probabilities are: </li></ul>
- 74. 10.2 Initial Problem Solution <ul><li>A friend who likes to gamble wagers that if you toss a coin repeatedly you will get 2 tails before you get 3 heads. Should you take the bet? </li></ul><ul><li>You can figure out what you should do by creating a probability tree diagram. </li></ul>
- 75. Initial Problem Solution, cont’d
- 76. Initial Problem Solution <ul><li>Add the probabilities of the outcomes that result in a win for you . </li></ul><ul><ul><li>You are much more likely to lose than to win, so you probably should not take the bet. </li></ul></ul>
- 77. Section 10.3 Conditional Probability, Expected Value, and Odds <ul><li>Goals </li></ul><ul><ul><li>Study conditional probability </li></ul></ul><ul><ul><ul><li>Study independent events </li></ul></ul></ul><ul><ul><li>Study odds </li></ul></ul>
- 78. 10.3 Initial Problem <ul><li>If you bet $100 on one number on the roulette wheel, what is your expected gain or loss? </li></ul><ul><ul><li>The solution will be given at the end of the section. </li></ul></ul>
- 79. Conditional Probability <ul><li>In the experiment of tossing 3 fair coins suppose you know the first coin came up heads. </li></ul><ul><ul><li>The sample space is {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. </li></ul></ul><ul><ul><li>Then the conditional sample space is {HHH, HHT, HTH, HTT} because the first coin was Head. </li></ul></ul>
- 80. Conditional Probability <ul><li>The probability of A given B is called a conditional probability . </li></ul><ul><li>The probability of A given B means the probability of event A occurring within the conditional sample space of event B . </li></ul><ul><ul><li>The probability of A given B is written P( A | B ), </li></ul></ul>
- 81. Conditional Probability <ul><li>Suppose A and B are events in a sample space S and that the probability of B is not zero. </li></ul><ul><li>The formula for conditional probability is </li></ul>
- 82. Example 1 <ul><li>One jar of marbles contains 2 white marbles and 1 black marble. </li></ul><ul><li>Another jar of marbles contains 1 white marble and 2 black marbles. </li></ul>
- 83. Example 1 <ul><li>A coin is tossed. </li></ul><ul><ul><li>Heads: a marble is selected from the first jar. </li></ul></ul><ul><ul><li>Tails: a marble is selected from the second jar. </li></ul></ul><ul><li>Find the probability that the coin landed heads up, given that a black marble was drawn. </li></ul>
- 84. Example 1 <ul><li>A coin is tossed. </li></ul><ul><ul><li>Heads: a marble is selected from the first jar. </li></ul></ul><ul><ul><li>Tails: a marble is selected from the second jar. </li></ul></ul><ul><li>Find the probability that the coin landed heads up, given that a black marble was drawn. </li></ul><ul><li>Solution: Create a probability tree diagram. </li></ul><ul><ul><li>The sample space is {HW, HB, TW, TB}. </li></ul></ul><ul><ul><li>The probabilities are labeled on the diagram. </li></ul></ul>
- 85. Example 1 <ul><li>Solution, cont’d: </li></ul>
- 86. Example 2 <ul><li>Suppose a test for a viral infection is not 100% accurate. </li></ul><ul><ul><li>Of the population, ¼ is infected and ¾ is not. </li></ul></ul><ul><ul><li>Of those infected, 90% test positive. </li></ul></ul><ul><ul><li>Of those not infected, 80% test negative. </li></ul></ul><ul><li>What is the probability the test is correct? </li></ul><ul><li>Given that a person’s test is positive, what is the probability the person is infected? </li></ul>
- 87. Example 2 <ul><li>Solution: Create a probability tree diagram. </li></ul><ul><ul><li>Of the population, ¼ is infected and ¾ is not. </li></ul></ul><ul><ul><li>Of those infected, 90% test positive. </li></ul></ul><ul><ul><li>Of those not infected, 80% test negative. </li></ul></ul>
- 88. Example 2 <ul><li>The probability of a correct test is the probability of a positive test for an infected person or a negative test for an uninfected person. </li></ul>
- 89. Example 2 <ul><li>Given that a person’s test is positive, what is the probability the person is infected? Find the conditional probability: </li></ul><ul><ul><li>P(positive) = </li></ul></ul>
- 90. Example 2 <ul><li>Solution, cont’d: </li></ul><ul><ul><li>P(infected and positive) = </li></ul></ul><ul><ul><li>P(infected|positive) </li></ul></ul><ul><ul><li>= </li></ul></ul>
- 91. Example 3 <ul><li>Results from an inspection of a candy company’s 2 production lines are shown in the table. </li></ul><ul><li>If a customer find a sub-standard piece of candy, what is the probability it came from the Bay City factory? </li></ul>
- 92. Example 3 <ul><li>Solution: Use the numbers in the table to solve: </li></ul><ul><ul><li>P(Bay City and sub-standard) = </li></ul></ul>
- 93. Example 3 <ul><li>Solution, cont’d: </li></ul><ul><ul><li>P(sub-standard) = </li></ul></ul>
- 94. Example 3 <ul><li>Solution, cont’d: </li></ul><ul><ul><li>P(Bay City|sub-standard) = </li></ul></ul>
- 95. Independent Events <ul><li>Two events are called independent if one event does not influence the other. </li></ul><ul><li>When 2 events are independent, their probabilities follow the rule given below: </li></ul>
- 96. Example 5 <ul><li>A student’s name is chosen at random from the college enrollment list and the student is interviewed. </li></ul><ul><ul><li>Let A be the event the student regularly eats breakfast. </li></ul></ul><ul><ul><li>Let B be the event the student has a 10:00 AM class. </li></ul></ul><ul><li>Explain in words what is meant by: </li></ul>
- 97. Example 5, <ul><li>Solution: </li></ul><ul><ul><li>: the probability the student regularly eats breakfast and has a 10:00 AM class. </li></ul></ul><ul><ul><li>: the probability the student regularly eats breakfast given that the student has a 10:00 AM class </li></ul></ul><ul><ul><li> : the probability the student does not regularly eat breakfast. </li></ul></ul>
- 98. Expected Value <ul><li>The average numerical outcome for many repetitions of an experiment is called the expected value . </li></ul><ul><ul><li>If E = 0, the game is said to be fair . </li></ul></ul>
- 99. Example 6 <ul><li>An experiment consists of rolling a fair die and noting the number on top of the die. </li></ul><ul><li>Compute the expected value of one roll of the die. </li></ul>
- 100. Example 6 <ul><li>Solution: The calculations are shown below: Multiply 1(1/6) and 2(1/6) and 3(1/6) etc. </li></ul>
- 101. Example 7 <ul><li>How much should an insurance company charge as its average premium in order to break even? </li></ul>
- 102. Example 7 <ul><li>Solution: The calculations are shown above. The product row is the product of the 2 numbers directly above each value in the product row. </li></ul><ul><ul><li>The average premium should cost $760. </li></ul></ul>
- 103. Odds <ul><li>The odds in favor of an event compare the number of favorable outcomes to the number of unfavorable outcomes. </li></ul><ul><ul><li>If the odds in favor are a : b , then </li></ul></ul>
- 104. Odds <ul><li>The odds against an event compare the number of unfavorable outcomes to the number of favorable outcomes. </li></ul><ul><ul><li>The odds in favor of an event E are </li></ul></ul><ul><ul><li>The odds against an event E are </li></ul></ul>
- 105. Example 8 <ul><li>Suppose a card is randomly drawn from a standard deck. </li></ul><ul><li>What are the odds in favor of drawing a face card? </li></ul>
- 106. Example 8 <ul><li>Solution: There are 12 face cards in the deck, and there are 40 other cards. </li></ul><ul><li>The odds in favor are 12:40, which simplifies to 3:10. </li></ul>
- 107. Example 9 <ul><li>Find P( E ) given the following odds: </li></ul><ul><ul><li>The odds in favor of E are 3:7. </li></ul></ul><ul><ul><li>The odds against E are 5:13. </li></ul></ul>
- 108. Example 9, cont’d <ul><li>Solution: </li></ul><ul><ul><li>The odds in favor of E are 3:7. </li></ul></ul><ul><ul><li>The odds against E are 5:13. </li></ul></ul>
- 109. Example 10 <ul><li>Find the odds in favor of event E , given the following probabilities: </li></ul><ul><ul><li>P( E ) = 1/4. </li></ul></ul><ul><ul><li>P( E ) = 3/5. </li></ul></ul>
- 110. Example 10, cont’d <ul><li>Solution: </li></ul><ul><ul><li>The odds in favor of E are 1:3. </li></ul></ul><ul><ul><li>The odds in favor of E are 3:2. </li></ul></ul>
- 111. 10.3 Initial Problem Solution <ul><li>A roulette wheel has 38 slots numbered 00, 0 and 1 through 36. You place a bet on a number or combination of numbers. If you bet on the winning number, you win your bet plus 35 times your bet. If you lose, you lose the money you bet. </li></ul><ul><li>If you bet $100 on one number, what is your expected gain or loss? </li></ul>
- 112. Initial Problem Solution <ul><li>Solution: </li></ul><ul><ul><li>The probability of winning is 1/38 and the amount won would be $3500. </li></ul></ul><ul><ul><li>The probability of losing is 37/38 and the amount lost would be $100. </li></ul></ul>
- 113. Initial Problem Solution <ul><li>The expected value is approximately -$5.26. You should expect to lose this amount, on average, for every $100 you bet. </li></ul>
- 114. Chapter 10 Assignments <ul><li>Section 10.1 pg 645 (7,11,15,17,31,35) </li></ul><ul><li>Section 10.2 pg 664 (1,3,11,13,19,25) </li></ul><ul><li>Section 10.3 pg 686 (5a,6a,15,27,3339,41) </li></ul>

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