Assist with the following questions;
13. More cards! Suppose you want to divide a 52 card deck into four hands with 13 cards each. 1What is the probability that each hand has a king? 14. Suppose you are taking a multiple-choice test with c choica for each question. In answering a question on this test, the probability that you know the answer is p. [f you don't know the answer, you cheese one at random. What is the probability that you knew the answer to a question, given that you answered it correctly? 15. ICorrupted by their p-L'iwer, the judges running the popular game show America's Nor! Top Molhenmtict'en have been taking bribm from many of the contestants. Each episode, a given contmtant is either allowed to stay on the show or is kicked off. If the contestant has been bribing the judges she will be allowed to stay with probability 1. [lithe contestant has not been bribing the judges, she wl be allowed to stay with probability 1:3. Suppose that 1,134 of the contestants have been bribing the judges. The same contestants bribe the judges in both rounds, i.e., if a contestant bribes them in the rst round, she bribes them in the second round toe (and vice verse}. {a} If you pick a random contestant who was allowed to stay during the rst episode, what is the probability that she was bribing the judges?I [b] If you pick a random contestant, what is the probability that she is allowed to stay during both of the first two episodes?I [c] If you pick random contestant who was allowed to stay during the rst episode, what is the probability that she gets kicked o' during the second episode? 16. Consider the Monty Hall problem. Let's label the door with the car behind it n and the other two doors b and c. In the game the contestant choosm a door and then Monty chooses a door, so we can label each outcome as *contestant followed by Monty\Rate of prostate cancer among men over 50 = 0.0005 True positive rate for the test = 0.9 False positive rate for the test = 0.01 Let T be the event a man has a positive test and let D be the event a man has a dangerous type of the disease. Find P(D ) and P(D T.). 19. A multiple choice exam has 4 choices for each question. A student has studied enough so that the probability they will know the answer to a question is 0.5, the probability that they will be able to eliminate one choice is 0.25, otherwise all 4 choices seem equally plausible. If they know the answer they will get the question right. If not they have to guess from the 3 or 4 choices. As the teacher you want the test to measure what the student knows. If the student answers a question correctly what's the probability they knew the answer? 20. Suppose you have an urn containing 7 red and 3 blue balls. You draw three balls at random. On each draw, if the ball is red you set it aside and if the ball is blue you put it22. Suppose that P(A) = 0.4, P(B) = 0.3 and P((A UB)") = 0.42. Are A and B independent? 23. Suppose now that events A, B and C are mutually independent with P(A) = 0.3, P(B) = 0.4, P(C) =0.5. Compute the following: (Hint: Use a Venn diagram) (i) P(AnBnC.) (ii) P(AnBond) (iii) P(AnBAC) 24. You roll a twenty-sided die. Determine whether the following pairs of events are independent. (a) 'You roll an even number' and 'You roll a number less than or equal to 10'. (b) 'You roll an even number' and 'You roll a prime number'. 25. Suppose A and B are events with 0
