4-3 Complements, Conditional Probability, and Bayes’ Theorem 177 The importance and usefulness of Bayes’ theorem is that it can be used with sequential events, whereby new additional information is obtained for a subsequent event, and that new information is used to revise the probability of the initial event. In this context, the terms prior probability and posterior probability are commonly used. DEFINITIONS A prior probability is an initial probability value originally obtained before any additional information is obtained. A posterior probability is a probability value that has been revised by using additional information that is later obtained. Relative to Example 4, P1C2 = 0.01, which is the probability that a randomly selected subject has cancer. P1C2 is an example of a prior probability. Using the additional information that the subject has received a positive test result, we found that P1C positive test result2 = 0.0748, and this is a posterior probability because it uses that additional information of the positive test result. Statistical Literacy and Critical Thinking 1.Language: Complement of “At Least One” Let A = the event of getting at least one defective calculator when four are randomly selected with replacement from a batch. Write a statement describing event A. 2.Probability of At Least One Let A = the event of getting at least 1 malfunctioning iPhone when 3 iPhones are randomly selected with replacement from a batch. Based on data from Statista, the malfunction rate is 7.5%. Which of the following are correct? a. P1A2 = 10.925210.925210.9252 = 0.791 b. P1A2 = 1 - 10.925210.925210.9252 = 0.209 c. P1A2 = 10.075210.075210.0752 = 0.000422 3. Notation For a polygraph (lie detector) used when a subject is presented with a question, let L = the subject lied and let Y = the polygraph indicated that the subject told a lie. Use your own words to translate the notation P1Y0 L2 into a verbal statement. 4.Confusion of the Inverse Using the same events L and Y described in Exercise 3, describe confusion of the inverse. At Least One. In Exercises 5–12, find the probability. 5. Four Girls Find the probability that when a couple has four children, at least one of them is a girl. (Assume that boys and girls are equally likely.) 6.Probability of a Girl Assuming that boys and girls are equally likely, find the probability of a couple having a boy when their third child is born, given that the first two children were both girls. 7. Births in the United States In the United States, the true probability of a baby being a boy is 0.512 (based on the data available at this writing). For a family having three children, find the following. a. The probability that the first child is a girl. b. The probability that all three children are boys. 4-3 Basic Skills and Concepts continued
RkJQdWJsaXNoZXIy NjM5ODQ=