5-2 Binomial Probability Distributions 219 PART 1 Basics of Binomial Probability Distribution Binomial probability distributions allow us to deal with circumstances in which the outcomes belong to two categories, such as heads>tails or acceptable>defective or survived>died. e or Go Figure 9,000,000: Number of other people with the same birthday as you. DEFINITION A binomial probability distribution results from a procedure that meets these four requirements: 1. The procedure has a fixed number of trials. (A trial is a single observation.) 2. The trials must be independent, meaning that the outcome of any individual trial doesn’t affect the probabilities in the other trials. 3. Each trial must have all outcomes classified into exactly two categories, commonly referred to as success and failure (but a “success” is not necessarily something good). 4. The probability of a success remains the same in all trials. Notation for Binomial Probability Distributions S and F (success and failure) denote the two possible categories of all outcomes. P1S2 = p 1p = probability of a success2 P1F2 = 1 - p = q 1q = probability of a failure2 n the fixed number of trials x a specific number of successes in n trials, so x can be any whole number between 0 and n, inclusive p probability of success in one of the n trials q probability of failure in one of the n trials P1x2 probability of getting exactly x successes among the n trials CAUTION The word success as used here is arbitrary and does not necessarily represent something good. Either of the two possible categories may be called the success S as long as its probability is identified as p. (The value of q can always be found from q = 1 - p. If p = 0.95, thenq = 1 - 0.95 = 0.05.) CAUTION When using a binomial probability distribution, always be sure that x and p are consistent in the sense that they both refer to the same category being called a success. Cash EXAMPLE 1 When an adult smartphone owner is randomly selected (with replacement), there is a 0.05 probability that this person is cashless (never carries cash) (based on a U.S. Bank survey of over 2003 adult smartphone owners in the U.S.). We want to find the probability that among ten randomly selected adults, exactly two of them are cashless. a. Does this procedure result in a binomial distribution? b. If this procedure does result in a binomial distribution, identify the values of n, x, p, and q. continued
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