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www.downloadslide.com 3.4 The Binomial Probability Distribution 101 questioned prior to a local election either favors candidate Jones or does not. In this section we are concerned **with** experiments, known as binomial experiments, that exhibit the following characteristics. DEFINITION 3.6 A binomial experiment possesses the following properties: 1. The experiment consists of a fixed number, n, of identical trials. 2. Each trial results in one of two outcomes: success, S, or failure, F. 3. The probability of success on a single trial is equal to some value p and remains the same from trial to trial. The probability of a failure is equal to q = (1 − p). 4. The trials are independent. 5. The random variable of interest is Y , the number of successes observed during the n trials. Determining whether a particular experiment is a binomial experiment requires examining the experiment for each of the characteristics just listed. Notice that the random variable of interest is the number of successes observed in the n trials. It is important to realize that a success is not necessarily “good” in the everyday sense of the word. In our discussions, success is merely a name for one of the two possible outcomes on a single trial of an experiment. EXAMPLE 3.5 Solution An early-warning detection system for aircraft consists of four identical radar units operating independently of one another. Suppose that each has a probability of .95 of detecting an intruding aircraft. When an intruding aircraft enters the scene, the random variable of interest is Y , the number of radar units that do not detect the plane. Is this a binomial experiment? To decide whether this is a binomial experiment, we must determine whether each of the five requirements in Definition 3.6 is met. Notice that the random variable of interest is Y , the number of radar units that do not detect an aircraft. The random variable of interest in a binomial experiment is always the number of successes; consequently, the present experiment can be binomial only if we call the event do not detect a success. We now examine the experiment for the five characteristics of the binomial experiment. 1. The experiment involves four identical trials. Each trial consists of determining whether (or not) a particular radar unit detects the aircraft. 2. Each trial results in one of two outcomes. Because the random variable of interest is the number of successes, S denotes that the aircraft was not detected, and F denotes that it was detected. 3. Because all the radar units detect aircraft **with** equal probability, the probability of an S on each trial is the same, and p = P(S) = P(do not detect) = .05.

www.downloadslide.com 102 Chapter 3 Discrete Random Variables and Their Probability Distributions 4. The trials are independent because the units operate independently. 5. The random variable of interest is Y , the number of successes in four trials. Thus, the experiment is a binomial experiment, **with** n = 4, p = .05, and q = 1 − .05 = .95. EXAMPLE 3.6 Solution Suppose that 40% of a large population of registered voters favor candidate Jones. A random sample of n = 10 voters will be selected, and Y , the number favoring Jones, is to be observed. Does this experiment meet the requirements of a binomial experiment? If each of the ten people is selected at random from the population, then we have ten nearly identical trials, **with** each trial resulting in a person either favoring Jones (S) or not favoring Jones (F). The random variable of interest is then the number of successes in the ten trials. For the first person selected, the probability of favoring Jones (S) is .4. But what can be said about the unconditional probability that the second person will favor Jones? In Exercise 3.35 you will show that unconditionally the probability that the second person favors Jones is also .4. Thus, the probability of a success S stays the same from trial to trial. However, the conditional probability of a success on later trials depends on the number of successes in the previous trials. If the population of voters is large, removal of one person will not substantially change the fraction of voters favoring Jones, and the conditional probability that the second person favors Jones will be very close to .4. In general, if the population is large and the sample size is relatively small, the conditional probability of success on a later trial given the number of successes on the previous trials will stay approximately the same regardless of the outcomes on previous trials. Thus, the trials will be approximately independent and so sampling problems of this type are approximately binomial. If the sample size in Example 3.6 was large relative to the population size (say, 10% of the population), the conditional probability of selecting a supporter of Jones on a later selection would be significantly altered by the preferences of persons selected earlier in the experiment, and the experiment would not be binomial. The hypergeometric probability distribution, the topic of Section 3.7, is the appropriate probability model to be used when the sample size is large relative to the population size. You may wish to refine your ability to identify binomial experiments by reexamining the exercises at the end of Chapter 2. Several of the experiments in those exercises are binomial or approximately binomial experiments. The binomial probability distribution p(y) can be derived by applying the samplepoint approach to find the probability that the experiment yields y successes. Each sample point in the sample space can be characterized by an n-tuple involving the

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