Please offer step-by-step answers.

1. We formulate a hypothesis. Our hypothesis is T (true) or F (false). Our prior belief is that the hypothesis is true with probability IF' [T] We design a test whose outcome is either 1 (conrming) or 0 (contradicting). llD [1|F] is the probability that the test conrms our hypothesis when the hypothesis is actually false (type I error). IF' [0|T] is the probability that the test contradicts our hypothesis when our hypothesis is actually true (type II error). (a) [1 point] If the test contradicts our hypothesis (i.e. returns 0), what is the probability that our hypothesis is true as a function of the prior belief P [T], the probability of making a type I error lF' [1|F], and the probability of making a type II error If\" [0|T]? (b) [2 points] Suppose that the hypothesis is actually false and the test is carried out independently N times. What is the probability that at least one repetition conrms the hypothesis (i.e. returns 1) as a function of the number of repetitions N and the probability of making a type I error IF\" [1|F]? How does this probability change with N