The prior probabilities for events A1 and A2 are P(A1) = 0.30 and P(A2) = 0.70. It is also known that P(A1 A2) = 0. Suppose P(B|A1) = 0.20 and P(B|A2) = 0.80. a. Are events A1 and A2 mutually exclusive? Yes v Explain. (i) P(A1 n A2) = 0 (ii) P(A1) + P042) = 1 (iii) P(A2) * P (A2 | A1) (iv) P(A2) 4* P(A2 | A1) - Select your answer - V b. Compute P(A1 F) B) (to 2 decimals). Compute P(A2 n B) (to 2 decimals). .14 c. Compute P(B) (to 2 decimals). d. Apply Bayes' theorem to compute P(A1|B) (to 4 decimals). Also apply Bayes' theorem to compute P(A2|B) (to 4 decimals). The prior probabilities for events A1, A2, and A3 are P(A1 ) = 0.20, P(A2 ) = 0.50, and P(A3 ) = 0.30. The conditional probabilities of event B given A1, A2 , and A3 are P(B|Al) = 0.60, P(B|A2) = 0.40, and P(B| A3) = 0.25. Round your answers to two decimal places. a. Compute P(Bn Al) , P(Bn A2), and P(Bn A3). P(Bn Al) = P(Bn A2) = P(Bn A3) = b. Apply Bayes' theorem, to compute the posterior probability P(A2 | B). P(A;) P(B| Ai) P(AinB ) = P(A ) P(B Al ) + P(A2) P(B| A2 ) +...+ P(An) P(B|An) c. Use the tabular approach to applying Bayes' theorem to compute P(Al | B), P(A2 | B), and P(As | B). Events P(Ai) P(B|Ai) P(Ain B) P(A: | B) A1 A2 A3 Total:A consulting rm submitted a bid for a large research project. The firm's management initially felt they had a 50- 50 chance of getting the project. However, the agency to which the bid was submitted subsequently requested additional information on the bid. Past experience indicates that for 73% of the successful bids and 38% of the unsuccessful bids the agency requested additional information. a. What is the prior probability of the bid being successful (that is, prior to the request for additional information) (to 1 decimal)? b. What is the conditional probability of a request for additional information given that the bid will ultimately be successful (to 2 decimals)? c. Compute the posterior probability that the bid will be successful given a request for additional information (to 2 decimals). An oil company purchased an option on land in Alaska. Preliminary geologic studies assigned the following prior probabilities. P(high-quality oil) 2 0.55 P(medium-qua.lity oil) = 0.25 P(no oil) 2 0.20 a. What is the probability of finding oil (to 2 decimals)? .a@ b. After 200 feet of drilling on the first well, a soil test is taken. The probabilities of finding the particular type of soil identified by the test are given below. P(soi1|highqua.lity oil) 2 0.15 P(soi1|mediumqua.lity oil) = 0.75 P(soil|no oil) 2 0.15 Given the soil found in the test, use Bayes' theorem to compute the following revised probabilities (to 4 decimals). P(high-quality oi1|soi1) .3667 0 P(mediumqua.lityoi1|soil) .5 0 P(no oil|soil) .1333 0 What is the new probability of nding oil (to 4 decimals)? .375 0 According to the revised probabilities, what is the quality of oil that is most likely to be found? @