Question: Corrupted by their power, the judges running the popular game show America's Next Top Mathematician have been taking bribes from many of the contestants. Each episode, a given contestant 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. If the contestant has not been bribing the judges, she will be allowed to stay with probability 1/3.
Suppose that 1/4 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 first round, she bribes them in the second round too (and vice versa).
a. If you pick a random contestant who was allowed to stay during the first episode, what is the probability that she was bribing the judges?
b. If you pick a random contestant, what is the probability that she is allowed to stay during both of the first two episodes?
c. If you pick random contestant who was allowed to stay during the first episode, what is the probability that she gets kicked off during the second episode?