Topic: Bayesian inference. Hypothesis test between two coins

Alioe has two coins. The probability oi | leads ior the iirst coin is lf'l, and the probability of l leads for the second is 3f4. Other than this diiierencc. the coins are indistinguishable. Alice chooses one of the coins at random and sends it to Bob. The random selection used by Alice to pick the coin to send to Bob is such that the first coin has a probabilityp oi being selected. Assume that c: p f. 1. Bob tries to guess which of the two coins he received by tossing it 3 times in a row and observing the outcome. Assume that for any particular coin_. all tosses of that coin are independent. 1. Given that Bob observed is l leads out ofthe 3 tosses {where k ID, 11 2, 3], what is the conditional probability that he received the first coin? E.-.) 3k '3] 334-13: l 3k-(l p] |:\\l 33): _ (1 _ P) aa-t-mat-(l p] 2. We define an error to have occurred if Bob decides that he received one coin from Alice, but he actually received the other coin. lle decides that he received the iirst coin when the number of l leadsr k, that he observes on thc3 tosses satisfies a certain condition. When one of the iollowing conditions is used, Bob will minimize the probability of error. Choose the correct threshold condition. ('1) none of the above 3_ For this part, assume that p = 33%. . What is the probability that Bob will guess the coin correctly using the decision rule irom part 2? . Suppose instead that Bob tries to guess which coin he received without tossing it. He still guesses the coin in order to minimize the probability oi error. What is the probability that Bob will guess the coin correctly under this scenario? 4. Bob uses the decision rule of Part 2. lip is small, then Bob will always decide in favor oi the second coinI ignoring the results oi the three tosses. The range of such p's is [I], 3). Find t