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Chapter 11 Probability Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 1 Chapter 11: Probability 11.1 11.2 11.3 11.4 11.5 Basic Concepts Events Involving “Not” and “Or” Conditional Probability and Events Involving “And” Binomial Probability Expected Value and Simulation Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 2 Section 11-1 Basic Concepts Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 3 Basic Concepts • Understand the basic terms in the language of probability. • Work simple problems involving theoretical and empirical probability. • Understand the law of large numbers (law of averages). • Find probabilities related to flower colors as described by Mendel in his genetics research. • Determine the odds in favor of an event and the odds against an event. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 4 Probability The study of probability is concerned with random phenomena. Even though we cannot be certain whether a given result will occur, we often can obtain a good measure of its likelihood, or probability. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 5 Probability In the study of probability, any observation, or measurement, of a random phenomenon is an experiment. The possible results of the experiment are called outcomes, and the set of all possible outcomes is called the sample space. Usually we are interested in some particular collection of the possible outcomes. Any such subset of the sample space is called an event. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 6 Example: Finding Probability When Tossing a Coin If a single fair coin is tossed, find the probability that it will land heads up. Solution The sample space S = {h, t}, and the event whose probability we seek is E = {h}. P(heads) = P(E) = 1/2. Since no coin flipping was actually involved, the desired probability was obtained theoretically. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 7 Theoretical Probability Formula If all outcomes in a sample space S are equally likely, and E is an event within that sample space, then the theoretical probability of the event E is given by number of favorable outcomes n( E ) P( E ) . total number of outcomes n( S ) Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 8 Example: Flipping a Cup A cup is flipped 100 times. It lands on its side 84 times, on its bottom 6 times, and on its top 10 times. Find the probability that it will land on its top. Solution From the experiment it appears that P(top) = 10/100 = 1/10. This is an example of experimental, or empirical, probability. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 9 Empirical Probability Formula If E is an event that may happen when an experiment is performed, then the empirical probability of event E is given by number of times event E occurred P( E ) . number of times the experiment was performed Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 10 Example: Finding Probability When Dealing Cards There are 2,598,960 possible hands in poker. If there are 36 possible ways to have a straight flush, find the probability of being dealt a straight flush. Solution 36 P(straight flush) .0000139 2,598,960 Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 11 Example: Gender of a Student A school has 820 male students and 835 female students. If a student from the school is selected at random, what is the probability that the student would be a female? Solution number of female students P(female) total number of students 835 0.505 820 + 835 Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 12 The Law of Large Numbers As an experiment is repeated more and more times, the proportion of outcomes favorable to any particular event will tend to come closer and closer to the theoretical probability of that event. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 13 Comparing Empirical and Theoretical Probabilities A series of repeated experiments provides an empirical probability for an event, which, by inductive reasoning, is an estimate of the event’s theoretical probability. Increasing the number of repetitions increases the reliability of the estimate. Likewise, an established theoretical probability for an event enables us, by deductive reasoning, to predict the proportion of times the event will occur in a series of repeated experiments. The prediction should be more accurate for larger numbers of repetitions. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 14 Probability in Genetics Gregor Mendel, an Austrian monk, used the idea of randomness to establish the study of genetics. To study the flower color of certain pea plants, he found that: Pure red crossed with pure white produces red. Mendel theorized that red is “dominant” (symbolized by R), while white is recessive (symbolized by r). The pure red parent carried only genes for red (R), and the pure white parent carried only genes for white (r). Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 15 Probability in Genetics Every offspring receives one gene from each parent which leads to the tables below. Every second generation is red because R is dominant. 1st to 2nd Generation offspring First Parent R R Second Parent r r Rr Rr Rr Rr 2nd to 3rd Generation offspring First Parent Copyright © 2016, 2012, and 2008 Pearson Education, Inc. R r Second Parent R r RR Rr rR rr 16 Example: Probability of Flower Color Referring to the 2nd to 3rd generation table (previous slide), determine the probability that a third generation will be a) red b) white Base the probability on the sample space of equally likely outcomes: S = {RR, Rr, rR, rr}. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 17 Example: Probability of Flower Color Solution S = {RR, Rr, rR, rr}. a) Since red dominates white, any combination with R will be red. Three out of four have an R, so P(red) = 3/4. b) Only one combination rr has no gene for red, so P(white) = 1/4. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 18 Odds Odds compare the number of favorable outcomes to the number of unfavorable outcomes. Odds are commonly quoted in horse racing, lotteries, and most other gambling situations. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 19 Odds If all outcomes in a sample space are equally likely, a of them are favorable to the event E, and the remaining b outcomes are unfavorable to E, then the odds in favor of E are a to b, and the odds against E are b to a. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 20 Example: Finding the Odds of Winning a TV 200 tickets were sold for a drawing to win a new television. If Matt purchased 10 of the tickets, what are the odds in favor of Matt’s winning the television? Solution Matt has 10 chances to win and 190 chances to lose. The odds in favor of winning are 10 to 190, or 1 to 19. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 21 Example: Converting from Probability to Odds Suppose the probability of rain today is 43%. Give this information in terms of odds. Solution 43 We can say that P (rain) .43 . 100 43 out of 100 outcomes are favorable, so 100 – 43 = 57 are unfavorable. The odds in favor of rain are 43 to 57 and the odds against rain are 57 to 43. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 22 Example: Converting from Odds to Probability Your odds of completing a College Algebra class are 16 to 9. What is the probability that you will complete the class? Solution There are 16 favorable outcomes and 9 unfavorable. This gives 25 possible outcomes. So 16 P(completion) 0.64. 25 Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 23