answer all questions with explanation

Show YOUR WORK(NO CURSIVE) 5a) In a casino in Los Vegas there are two slot machines that each take $1 and will either pay out $10 (win) or nothing (lose). They are identical, except that one machine has been set such that you will win 10% of the time, while the other is more generous - it has been set such that you will win 209% of the time. Obviously, you would like to play on the more generous (20%) machine, but you do not know which machine is which. You adopt the following strategy: you assume initially that the two machines are equally likely to be the more generous (20% win) machine. You then select one of the two machines at random and put a coin into it. Suppose that you lose: use that fact to estimate the probability that the machine you selected is the more generous of the two machines. (Hint: use Bayes' theorem!) Event A Is: Event B Is: P(BIA) = P(A) = P(B) = P(AIB) = 5b) What if you play a second time on the same machine and lose again: what is the probability now? You can compute this 2 ways. First, you can make P(B) the chances of losing two times in a row, and then apply Bayes' rule. (Hint: to compute the probability of losing twice in a row, look back at Question 3c.) Event A Is: Event B Is: Losing two times in a row P(BIA) = P(A) = P(B) = P(AIB) = 5c) Alternatively, you can use the results of the first computation to update your priors, and then apply Bayes' using only the results of the second bet. Do you get the same answer? (Hint: you should!) Event A Is: Event B Is: P(BIA) = P(A) = P(B) = P(AIB) = 5d) Now consider a machine that pays off 109% of the time versus another that pays off 90% of the time. If you lose your first bet, what is the probability that the machine you selected is the more generous of the two? P(AIB) = 6) The Monte Hall Problem: You are on a game show, and given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then asks, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice? More specifically, 6a) What is the probability of winning the car if you switch? 6b) What is the probability if you do not switch