15. 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?

15. Suppeae tbat buses arrive are scheduled to arrive at a bus stop at noon but are always I minutes late 1wl'uere X is an exponential random variable with probability density function fx{z} = led\". Suppeoe that you arrive at the bus stop preeiaely at noon. [a] Compute the probability that you have to wait for more than Ewe minutes for the bus to arrive. [11] Suppose that you have already waiting for 1 minutes. Compute the probability that you have to wait an additional litre minute; or more. \f\f