Section 7.6 Bayes Theorem

Transcription

1 Section 7.6 Bayes Theorem Bayes Theorem Let A 1, A 2,, A n be a partition of a sample space S, and let E be an event of the experiment such that P (E) 0 and P (A i ) 0 for 1 i n. Then the conditional probability P (A i E) (1 i n) is given by P (A i E) = P (A i ) P (E A i ) P (A 1 ) P (E A 1 ) + P (A 2 ) P (E A 2 ) + + P (A n ) P (E A n ) Recall from section 7.5 that P (A i E) = P (A i ) P (E A i ). Also, P (E) = P (A 1 ) P (E A 1 ) + P (A 2 ) P (E A 2 ) + + P (A n ) P (E A n ). Therefore, we could use the conditional probability rule from section 7.5 and say that P (A i E) = P (A i E) P (E) 1. Find P (F B) and P (E A) using the tree diagram. (Round answers to three decimal places.)

2 2. A survey involving 700 likely Democratic voters and 200 likely Republican voters asked the question: Do you support or oppose legislation that would require trigger locks on guns, to prevent misuse by children? The following results were obtained: Answer Democrats, % Republicans, % Support Oppose 8 17 Don t know/refused 7 10 If a randomly chosen respondent in the survey answered support, what is the probability that he or she is a likely Republican voter? (Round answer to three decimal places.) 3. Applicants who wish to be admitted to a certain professional school in a large university are required to take a screening test devised by an educational testing service. From past results, the testing service has established that 70% of all applicants are eligible for admission and that 90% of those who are eligible for admission pass the exam, whereas 14% of those who are ineligible for admission pass the exam. (Round answers to three decimal places.) 2 Fall 2017, Maya Johnson

3 (a) What is the probability that an applicant for admission passed the exam? (b) What is the probability that an applicant for admission who passed the exam was actually ineligible? 4. There are three jars that each contain 10 marbles. The first contains 3 white marbles and 7 red marbles, the second 6 white and 4 red, and the third all 10 white. An experiment consists of first selecting a jar at random. (Assume each jar has an equal probability of being selected.) After a jar is selected, a marble is randomly drawn from this jar, noting its color. If the marble drawn was white, find the probability that the third jar was selected. (Round answer to three decimal places.) 3 Fall 2017, Maya Johnson

4 5. The Office of Admissions and Records of a large western university released the accompanying information concerning the contemplated majors of its freshman class. (Round answers to three decimal places.) % of Freshmen % of Major % of Major Choosing That is That is Major This Major Female Male Business Humanities Education Social science Natural sciences Other (a) What is the probability that a student selected at random from the freshman class is a female? (b) What is the probability that a business student selected at random from the freshman class is a male? (c) What is the probability that a female student selected at random from the freshman class is majoring in business? 4 Fall 2017, Maya Johnson

5 6. Three machines turn out all the products in a factory, with the first machine producing 35% of the products, the second machine 25%, and the third machine 40%. The first machine produces defective products 6% of the time, the second machine 17% of the time and the third machine 4% of the time. What is the probability that a non-defective product came from the second machine? (Round answer to four decimal places.) 7. Box A contains seven white marbles and five black marbles. Box B contains six white marbles and four black marbles. An experiment consists of first selecting a marble at random from Box A. The marble is transferred to Box B and then a second marble is drawn from Box B. What is the probability that the first marble was white given that the second marble was white? (Round answer to three decimal places.) 5 Fall 2017, Maya Johnson

6 8. A medical test has been designed to detect the presence of a certain disease. Among people who have the disease, the probability that the disease will be detected by the test is However, among those who do not have the disease, the probability that the test will detect the presence of the disease is It is estimated that 3% of the population who take this test actually have the disease. (Round answers to three decimal places.) (a) If the test administered to an individual is positive (the disease is detected), what is the probability that the person actually has the disease? (b) If the test administered to an individual is negative (the disease is not detected), what is the probability that the person actually does have the disease? 6 Fall 2017, Maya Johnson