202 CHAPTER 4 Discrete Probability Distributions Identifying and Understanding Binomial Experiments Determine whether each experiment is a binomial experiment. If it is, specify the values of n, p, and q, and list the possible values of the random variable x. If it is not, explain why. 1. A certain surgical procedure has an 85% chance of success. A doctor performs the procedure on eight patients. The random variable represents the number of successful surgeries. 2. A jar contains five red marbles, nine blue marbles, and six green marbles. You randomly select three marbles from the jar, without replacement. The random variable represents the number of red marbles. SOLUTION 1. The experiment is a binomial experiment because it satisfies the four conditions of a binomial experiment. In the experiment, each surgery represents one trial. There are eight surgeries, and each surgery is independent of the others. There are only two possible outcomes for each surgery—either the surgery is a success or it is a failure. Also, the probability of success for each surgery is 0.85. Finally, the random variable x represents the number of successful surgeries. n = 8 Number of trials p = 0.85 Probability of success q = 1 - 0.85 = 0.15 Probability of failure x = 0, 1, 2, 3, 4, 5, 6, 7, 8 Possible values of x 2. The experiment is not a binomial experiment because it does not satisfy all four conditions of a binomial experiment. In the experiment, each marble selection represents one trial, and selecting a red marble is a success. When the first marble is selected, the probability of success is 5/20. However, because the marble is not replaced, the probability of success for subsequent trials is no longer 5/20. So, the trials are not independent, and the probability of a success is not the same for each trial. TRY IT YOURSELF 1 Determine whether the experiment is a binomial experiment. If it is, specify the values of n, p, and q, and list the possible values of the random variable x. If it is not, explain why. A multiple-choice quiz consists of 10 questions. Each question has four possible answers, only one of which is correct. To complete the quiz, a student who did not study randomly guesses the answer to each question. The random variable represents the number of correct answers. Answer: Page A38 For a random sample collected without replacement, such as in a survey, the events are dependent. However, you can treat this situation as a binomial experiment by treating the events as independent when the sample size is no more than 5% of the population. That is, n … 0.05N. EXAMPLE 1 Picturing the World A survey of 2647 U.S. parents of a child age 11 and under was conducted to study the ways in which children use social media. One of the questions from the survey and the responses (either yes or no) are shown below. (Adapted from Pew Research Center) Survey question: As far as you know, does this child ever watch videos onYouTube? No 20% Yes 80% Why is this a binomial experiment? Identify the probability of success p. Identify the probability of failure q.
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