6. (ASV, Exercise 2.40) 3 points. Incoming students at a certain school take a mathematics placement exam. The possible scores are 1, 2, 3, and 4. From past experience, the school knows that if a particular student's score is a: E {1,2,3,4}, then the student will become a mathematics major with probability %. Suppose that the incoming class had the following scores: 10% of the students scored a 1, 20% of the students scored a 2, 60% scored a 3, and 10% scored a 4. (a) What is the probability that a randomly selected student from the incoming class will become a mathematics major? (b) Suppose a randomly selected student from the incoming class turns out to be a mathematics major. What is the probability that he or she scored a 4 on the placement exam? 7. (ASV, Exercise 2.42) 2 points. Urn A contains 2 red and 4 white balls, and um B contains 1 red and 1 white ball. A ball is randomly chosen from urn A and put into urn B, and a ball is then chosen from urn B. What is the conditional probability that the transferred ball was white given that a white ball is selected from urn B? 8. 3 points. Consider the Monty Hall problem. In the case that the door you initially chose hid the valuable prize, suppose that Monty opens one of the other two doors for you not uniformly at random, but rather opens the second door with probability p and the third door with probability 1 p, for some p E [0, 1]. How does this affect your chances of getting the prize when switching