# Applied Math 40S Slides Feb 22, 2007

## by Darren Kuropatwa on Feb 22, 2007

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Introduction to probability. Basic definitions, applications and problem solving.

Introduction to probability. Basic definitions, applications and problem solving.

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## Applied Math 40S Slides Feb 22, 2007Presentation Transcript

• Experimental and Theoretical Probability In a family of three children, what is the probability that 2 of the children will be girls? Blaise Pascal Pierre de Fermat The Story of Probability The Chevalier de Mere, a rich Frenchman who liked gambling, was responsible for inviting the philosopher and mathematician Blaise Pascal to carry out some of the earliest work on probability theory. De Mere played a gambling game in which he bet that he could throw a six in four throws of a die. De Mere progressed from this game to betting that with two dice he could throw a double six in 24 throws. It was known that the odds were in his favour with the first game, and gamblers of the time reckoned that as four is to six (the numbers of ways a die can fall) as 24 is to 36 (the ways two dice can fall), the second game should be favourable. The Chevalier de Mere was not satisfied with this assumption and asked Pascal to work out the true probabilities. Source: http://www.probabilitytheory.info/topics/periodic_events.htm Photo source: http://ﬂickr.com/photos/phitar/7971517/
• Terms you should know ... DICTIONARY PROBABILITY: The branch of mathematics that deals with chance SAMPLE SPACE: The set of all possible things that can happen for a given set of circumstances Example: rolling a die-6, the sample space would be {1, 2, 3,4,5, 6} because these are all the possible outcomes. EVENT (E): An event is a subset of the sample space. It is one particular outcome for a given set of circumstances. SIMPLE EVENTS: The result of an experimental carried out in 1 step. Example: Flip a coin. The result is Heads. COMPOUND EVENT: The result of an experimental carried out in more than one step. Example: Flip a coin and roll a die. The result is heads and 6. Calculating the Probability of an Event Probability Can Be Expressed As: • a ratio • a fraction • a decimal • a percent CERTAIN EVENTS: An events whose probability is equal to 1. IMPOSIBLE EVENTS: An event whose probability is equal to 0. IMPORTANT: Probability is always a number between 0 and 1.
• Write your answers as fractions reduced to lowest terms. 1. What is the probability that a woman will win the Oscar Award for Best Actress? 2. What is the probability that a 7 shows when rolling a normal six-sided die? 3. What is the probability that a king is drawn from a normal deck of 52 cards? 4. A bag contains eight blue and five white marbles. What is the probability of randomly selecting a white marble? 5. A bag contains five red, four green, and three black candies. What is the probability that you do not select a black candy if you randomly select one?
• Complimentary Events The compliment of an event, E, is writtin as either E' or E. The compliment of an event refers to the case where E does not occur. Example: H = Drawing a heart from a deck of cards. H'(the compliment) = Drawing a card that is not a heart. Calculating complimentary probabilities ... P(E) + P(E') = 1 so ... P(E) = 1 – P(E') or P(E') = 1 – P(E) Understanding the concept ... If there are 52 players in a sudden death singles tennis tournament, how many games must be played in order to determine the winner?
• In a family of three children, what is the probability that 2 of the children will be girls? Theoretical Probability: The chances of events happening as determined by calculating results that would occur under ideal circumstances. For example, the theoretical probability of rolling a 4 on a four-sided die is 1/4 or 25%, because there is one chance in four to roll a 4, and under ideal circumstances one out of every four rolls would be a 4. Contrast this with experimental probability (next slide). DICTIONARY
• In a family of three children, what is the probability that 2 of the children will be girls? Experimental Probability: The chances of something happening, based on repeated testing and observing results. It is the ratio of the number of times an event occurred to the number of times tested. For example, to find the experimental probability of winning a game, one must play the game many times, then divide the number of games won by the total number of games played.
• Binomial Experiments & Probability DICTIONARY Simulating Binomial Experimets: randBin(# of trials, prob. of success, # of simulations) What is the probability of getting exactly 2 heads when fliping 3 coins 40 times? You'll need to know the the theoretical probability of this result is 3/8. Here is how to do the experiment on your calculator. Step Action 1. Press [Math] button on TI83 calculator. 2. Select [Prob]. 3. Select [randBin] (random binomial experiment). 4. Type in (1, 3/8, 40) . 1 represents the outcome for success (failure is 0). 3/8 represents the theoretical probability of success. 40 represents number of times the experiment is repeated. 5. Press enter and a result will show in row. 6. Press [STO] [2nd] [L1] to store the results in List 1. 7. Press [2nd] [STAT]. 8. Select [MATH] [Sum] [2nd] [L1] to find the sum of all the values in List 1. 9. Since "success = 1" and "failure = 0" this sum represents the number of successes. http://www.random.org/ You can also use this website: Random.org Photo source: http://ﬂickr.com/photos/marko_k/142582218/
• WHAT WOULD WE DO FOR 4O FAMILIES? you would have to change the 60 to 120 in order to cal. the number of babies in 40 families. I'm DONE! YOU WOULD ALSO HAVE TO CHANGE THE 20 TO A 40.
• h Pathways & Pascal's Triangle A B C How many ways can you get from point A to point B How many ways can you get from point A to point C assuming you always take the shortest route possible? assuming you always take the shortest route possible? You always take the shortest route possible to get from point A to Point C. Assuming you choose your route at random, what is the probability that you will pass through point B?
• The diagram below shows a game of chance where a ball is dropped as indicated, and eventually comes to rest in one of the four locations labelled A, B, C, or D. The ball is equally likely to go left or right each time it strikes a triangle. We want to determine the theoretical probability of a ball landing in any one of these four locations. To do this, we need to know the total number of paths the ball can take, and also the number of paths to each location.
• How many ways can the word RIVER be found in the array of letters shown to the right if you start from the top R and move diagonally down to the bottom R?
• Find the probability of flipping three pennies and getting at least 1 heads. have a good day everyone!!