Chapter 11 Probability

Chapter 11: Probability 11.1 Basic Concepts 11.2 Events Involving “Not” and “Or” 11.3 Conditional Probability and Events Involving “And” 11.4 Binomial Probability 11.5 Expected Value and Simulation

Section 11-1 p.584 Basic Concepts

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.

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. Deterministic phenomenon can be predicted exactly on the basis of obtainable information. Random phenomenon fluctuates in such a way that its value cannot be predicted exactly with obtainable information.

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.

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.

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

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.

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

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

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

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.

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.

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).

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 2nd to 3rd Generation Second Parent r First Parent R Rr Second Parent R r First Parent RR Rr rR rr offspring offspring

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}.

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.

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.

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.

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.

Example: Converting from Probability to Odds Suppose the probability of rain today is 43%. Give this information in terms of odds. Solution We can say that 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.

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