Baye's Rule:

P(B| A)p(A) Bayes' rule: P(A B) = p(B A)p(A)+p(B|~A)p(~A) 1. Suppose your friend presents you with 5 coins. She tells you that three of them are fair, meaning that if flipped, they will return heads or tails with 0.5 probability. She also mentions that two are weighted, and will return heads 0.75 of the time and tails 0.25. (a) You pick one at random and flip it; it returns heads. Calculate the probability that the coin you are holding is one of the weighted coins given that it returned heads on the first flip. (5 pts.) (b) Calculate the probability that the coin you are holding is a weighted one given that it returns heads on two out of two flips. (5 pts.) (c) Your friend informs you that you did indeed pick a weighted coin. You set it aside and pick another one at random from the remaining coins. You flip this new coin once and it returns tails. Calculate the probability that this new coin is weighted. (5 pts.)