M05_SULL8028_03_SE_C05.QXD 9/9/08 7:59 PM Page 257 Probability PART and Probability Distributions CHAPTER3 5 We now take a break from the statistical process. Why? In Probability Chapter 1, we mentioned that inferential statistics uses methods CHAPTER 6 that generalize results obtained from a sample to the population Discrete Probability Distributions and measures their reliability. But how can we measure their reliability? It turns out that the methods we use to generalize CHAPTER 7 The Normal results from a sample to a population are based on probability Probability and probability models. Probability is a measure of the likelihood Distribution that something occurs. This part of the course will focus on meth- ods for determining probabilities. M05_SULL8028_03_SE_C05.QXD 9/9/08 7:59 PM Page 258 5 Probability Outline 5.1 Probability Rules 5.2 The Addition Rule and Complements 5.3 Independence and the Have you ever watched a sporting Multiplication Rule event on television in which the an- 5.4 Conditional Probability nouncer cites an obscure statistic? and the General Where do these numbers Multiplication Rule come from? Well, pretend 5.5 Counting Techniques that you are the statisti- 5.6 Putting It Together: Which cian for your favorite Method Do I Use? sports team. Your job 5.7 Bayes’s Rule (on CD) is to compile strange or obscure probabili- ties regarding your favorite team and a competing team. See the Decisions project on page 326. PUTTING IT TOGETHER In Chapter 1, we learned the methods of collecting data. In Chapters 2 through 4, we learned how to summa- rize raw data using tables, graphs, and numbers. As far as the statistical process goes, we have discussed the collecting, organizing, and summarizing parts of the process. Before we can proceed with the analysis of data, we introduce probability, which forms the basis of in- ferential statistics. Why? Well, we can think of the probability of an outcome as the likelihood of observing that outcome. If something has a high likelihood of happening, it has a high probability (close to 1). If some- thing has a small chance of happening, it has a low probability (close to 0). For example, in rolling a single die, it is unlikely that we would roll five straight sixes, so this result has a low probability. In fact, the proba- bility of rolling five straight sixes is 0.0001286. So, if we were playing a game that entailed throwing a single die, and one of the players threw five sixes in a row, we would consider the player to be lucky (or a cheater) because it is such an unusual occurrence. Statisticians use probability in the same way. If something occurs that has a low probability, we investigate to find out “what’s up.” 258 M05_SULL8028_03_SE_C05.QXD 9/9/08 7:59 PM Page 259 Section 5.1 Probability Rules 259 5.1 PROBABILITY RULES Preparing for This Section Before getting started, review the following: • Relative frequency (Section 2.1, p. 68) Objectives 1 Apply the rules of probabilities 2 Compute and interpret probabilities using the empirical method 3 Compute and interpret probabilities using the classical method 4 Use simulation to obtain data based on probabilities 5 Recognize and interpret subjective probabilities Note to Instructor Probability is a measure of the likelihood of a random phenomenon or chance If you like, you can print out and distrib- behavior. Probability describes the long-term proportion with which a certain ute the Preparing for This Section quiz outcome will occur in situations with short-term uncertainty. located in the Instructor’s Resource The long-term predictability of chance behavior is best understood through a Center. The purpose of the quiz is to simple experiment. Flip a coin 100 times and compute the proportion of heads ob- verify that the students have the prereq- served after each toss of the coin. Suppose the first flip is tails, so the proportion of uisite knowledge for the section. 0 1 heads is ; the second flip is heads, so the proportion of heads is ; the third flip is Note to Instructor 1 2 2 Probabilities can be expressed as frac- heads, so the proportion of heads is ; and so on. Plot the proportion of heads versus tions, decimals, or percents. You may 3 want to give a brief review of how to con- the number of flips and obtain the graph in Figure 1(a). We repeat this experiment vert from one form to the other. with the results shown in Figure 1(b). Figure 1 1.0 1.0 0.9 0.9 3rd flip heads s s 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 2nd flip heads 0.3 0.3 Proportion of Head Proportion of Head 0.2 0.2 st 0.1 1 flip tails 0.1 0.0 0.0 0 50 100 0 50 100 Number of Flips Number of Flips (a) (b) In Other Words Looking at the graphs in Figures 1(a) and (b), we notice that in the short term Probability describes how likely it is that (fewer flips of the coin) the observed proportion of heads is different and unpredictable some event will happen. If we look at the for each experiment.As the number of flips of the coin increases, however, both graphs proportion of times an event has occurred tend toward a proportion of 0.5. This is the basic premise of probability. Probability over a long period of time (or over a large deals with experiments that yield random short-term results or outcomes yet reveal number of trials), we can be more certain long-term predictability. The long-term proportion with which a certain outcome is of the likelihood of its occurrence. observed is the probability of that outcome. So we say that the probability of observing 1 a head is or 50% or 0.5 because, as we flip the coin more times, the proportion of 2 1 heads tends toward . This phenomenon is referred to as the Law of Large Numbers. 2 The Law of Large Numbers As the number of repetitions of a probability experiment increases, the propor- tion with which a certain outcome is observed gets closer to the probability of the outcome. M05_SULL8028_03_SE_C05.QXD 9/9/08 7:59 PM Page 260 260 Chapter 5 Probability Note to Instructor The Law of Large Numbers is illustrated in Figure 1. For a few flips of the coin, In-class activity: Have each student flip a the proportion of heads fluctuates wildly around 0.5, but as the number of flips in- coin three times. Use the results to find creases, the proportion of heads settles down near 0.5. Jakob Bernoulli (a major con- the probability of a head. Repeat this tributor to the field of probability) believed that the Law of Large Numbers was experiment a few times. Use the cumula- common sense. This is evident in the following quote from his text Ars Conjectandi: tive results to illustrate the Law of Large “For even the most stupid of men, by some instinct of nature, by himself and without Numbers. You may also want to use the any instruction, is convinced that the more observations have been made, the less probability applet to demonstrate the Law of Large Numbers. See Problem 57. danger there is of wandering from one’s goal.” In probability, an experiment is any process with uncertain results that can be repeated.The result of any single trial of the experiment is not known ahead of time. However, the results of the experiment over many trials produce regular patterns that enable us to predict with remarkable accuracy. For example, an insurance com- pany cannot know ahead of time whether a particular 16-year-old driver will be in- volved in an accident over the course of a year. However, based on historical records, the company can be fairly certain that about three out of every ten 16-year- old male drivers will be involved in a traffic accident during the course of a year. Therefore, of the 816,000 male 16-year-old drivers (816,000 repetitions of the exper- iment), the insurance company is fairly confident that about 30%, or 244,800, of the drivers will be involved in an accident. This prediction forms the basis for establish- ing insurance rates for any particular 16-year-old male driver. We now introduce some terminology that we will need to study probability. Definitions The sample space, S, of a probability experiment is the collection of all possible outcomes. In Other Words An event is any collection of outcomes from a probability experiment.An event An outcome is the result of one trial may consist of one outcome or more than one outcome. We will denote events of a probability experiment. The sample with one outcome, sometimes called simple events,ei. In general, events are space is a list of all possible results of denoted using capital letters such as E. a probability experiment. The following example illustrates these definitions. EXAMPLE 1 Identifying Events and the Sample Space of a Probability Experiment Problem: A probability experiment consists of rolling a single fair die. (a) Identify the outcomes of the probability experiment. (b) Determine the sample space. A fair die is one in which each (c) Define the event E = “roll an even number.” possible outcome is equally likely. For example, rolling a 2 is just as likely as Approach: The outcomes are the possible results of the experiment.