1. Hypothesis test between two coins

Alice has two coins. The probability of Heads for the first coin is 1/4, and the probability of Heads for the second is 3/4. Other than this difference, the coins are indistinguishable. Alice chooses one of the coins at random and sends it to 80b. The random selection used by Alice to pick the coin to send to 80b is such that the first coin has a probability p of being selected. Assume that O < p < 1 . 80b 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 80b observed k Heads out of the 3 tosses (where k = 0, 1, what is the conditional probability that he received the first coin? o o o 33k 3k (I 33-k 33-k P 33 p) p+3k 33-k (1 -p) 33-k p+3k 2. We define an error to have occurred if 30b decides that he received one coin from Alice, but he actually received the other coin. He decides that he received the first coin when the number of Heads, k, that he observes on the 3 tosses satisfies a certain condition. When one of the following conditions is used, 30b will minimize the probability of error. Choose the correct threshold condition. k < log3 o k > loga o k logg o k 2 logg o none of the above