Problem 1 During an ultrasound check of a pregnant woman the ultrasound technician can try to determine the gender of the baby. When the baby turns out to be male (eventually1 when it. is born], during the ultrasolmd check, in 60% of the cases they say that the baby is a boy! in 211% of the cases they say that it is a girl and in 20% of the cases they say that the picture is inconclusive. If the baby turns out to be female1 in 50% of the cases they say it is a girl! in 10% of the cases they say that it is a boy, and in \"lib of the cases they say that the picture is inconclusive. Assume that the true probabilities of the baby being a boy or a girl are equal. Answer the following questions: 1.With what probability does the test say that the baby is a girl, a boy, or inconclusive? 2. The test says that the baby is a girl. With what probability is the baby a girl? 3. The test is inconclusive. With what probability is the baby a girl? 4. Show that the woman's beliefs after the test are a martingale (of her prior belief that the baby is a girl or a boy with probabilities g}. Problem 2 Let the state set be D. ={wmwl}. the action set beA = {email}. and the agent's utility function u (a, m} be What is an optimal decision as the function of the probability I = Pr (w = will? 1What 'm the expected utility as the function of this probability? Illustrate. Answer the same questions for the following matrix: Problem 3 Aygol's prior belief that she will get an 1&fo (C or less] in the nal exam are 31%;\" 50%,? 211%. If she is to get an 15.fo {C or less}, she solves all the problems in an assignment with probability 90%} 50%f 10% [conditionally independent from other as- signments]