Question

1. Use Bayes' theorem or a tree diagram to calculate the indicated probability. Round your answer to four decimal places. HINT [See Example 3.]

P(A | B) = .2, P(B) = .4, P(A | B') = .6. Find P(B | A).

P(B | A) =

2. Use Bayes' theorem or a tree diagram to calculate the indicated probability. Round your answer to four decimal places. HINT [See Example 3.]

P(X | Y) = 0.4, P(Y' ) = 0.3, P(X | Y' ) = 0.5. Find P(Y | X).

3. Use Bayes' theorem or a tree diagram to calculate the indicated probability. Round your answer to four decimal places. HINT [See Example 3.]

Y₁, Y₂, Y₃ form a partition of S. P(X | Y₁) = .7, P(X | Y₂) = .2, P(X | Y₃) = .9, P(Y₁) = .2, P(Y₂) = .3. Find P(Y₁ | X).

P(Y₁ | X) =

4. Suppose that it snows in Greenland an average of once every 27 days, and when it does, glaciers have a 20% chance of growing. When it does not snow in Greenland, glaciers have only a 1% chance of growing. What is the probability that it is snowing in Greenland when glaciers are growing? (Round your answer to four decimal places.)

5. In fall 2014, 34% of applicants with a Math SAT of 700 or more were admitted by a certain university, while 14% with a Math SAT of less than 700 were admitted. Further, 38% of all applicants had a Math SAT score of 700 or more. What percentage of admitted applicants had a Math SAT of 700 or more? (Round your answer to the nearest percentage point.)

6. In a survey in a certain year of married couples with earnings, 97% of all husbands were employed. Of all employed husbands, 76% of their wives were also employed. Noting that either the husband or wife in a couple with earnings had to be employed, find the probability that the husband of an employed woman was also employed. (Round your answer to four decimal places.)

7. In a certain year, 86% of all Caucasians in the U.S., 75% of all African-Americans, 75% of all Hispanics, and 76% of residents not classified into one of these groups used the Internet for e-mail. At that time, the U.S. population was 65% Caucasian, 11% African-American, and 14% Hispanic. What percentage of U.S. residents who used the Internet for e-mail were Hispanic? (Round your answer to the nearest whole percent.)

8. Any athlete who fails the Enormous State University's women's soccer fitness test is automatically dropped from the team. Last year, Mona Header failed the test, but claimed that this was due to the early hour. (The fitness test is traditionally given at 5 AM on a Sunday morning.) In fact, a study by the ESU Physical Education Department suggested that 51% of athletes fit enough to play on the team would fail the soccer test, although no unfit athlete could possibly pass the test. It also estimated that 40% of the athletes who take the test are fit enough to play soccer. Assuming these estimates are correct, what is the probability that Mona was justifiably dropped? (Round your answer to four decimal places.)

Answer 1

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P(A|B) = .2........1
P(B) = .4..........2
Using (1) and (2) :
P(A|B) = P(A and B)/P(B) = .2
P(A and B)/.4 = .2
P(A and B) = .08

P(A|B') = .6
P(A and B')/P(B') = .6
P(A and B') = .6*.6 = .36
P(A and B') = .36
P(A) = P(A and B) + P(A and B') = .08+.36 = .44
P(B|A) = P(A and B)/P(A) = .08/.44 = 0.1818

Answer: P(B|A) = 0.1818