8) (20 pts.) Corrupted by their power, the judges running the popular game show
ID: 3314745 • Letter: 8
Question
8) (20 pts.) Corrupted by their power, the judges running the popular game show Americas Next Top Scientist 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 staywith 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?Explanation / Answer
(a) Pr(stay l Bribe) = 1
Pr(stay l not bribed) = 1/3
Pr(Bribe) = 1/4
Pr(not bribe) = 3/4
so Pr(Probabiity tht a person reached second round she has bribed the judge) =
Pr(Bribe l Reached second round) = Pr(Bribe) * Pr(Second Round l Bribe) / [Pr(Bribe) * Pr(Second Round l Bribe) + Pr(Not bribe) * Pr(Second round l not bribe) ]
= 1/4 * 1/ [ 1/4 * 1 + 3/4 * 1/3]
= 1/4/ (1/2) = 1/2
(b) A random person is allowed to stay durin both of the first episodes.
So if the person who gives bribe, that means she will stay for both the two rounds.
If the person doesn't give bribes he has the probability to stay in the one round is 1/3.
Pr(A random person will stay during both of episodes) = Pr(Briber person)* Pr(to stay) + Pr(No bribe person) * Pr(stay in first round) * Pr(stay in secon round)
= 1/4 * 1 * 1 + 3/4 * 1/3 * 1/3 = 1/3
(c) Here a random person is llowed to stay during the first episode, the probability that she will get kicked off during the next episode.
She will be kicked only if she belong to nonbribe ctegory. In question (a), it is known that random person who stay during the first round, has 1/2 chance tht she bribed thejudges. So there is 1/2 chance too that she doesn't bribed te judges. So,
Pr(Non bribe person stayed in first round) = 1/2
Pr(Non bribe person kicked off from second round l she clered first round) = 1/2 * 1/3/ [ 1/2 * 1 + 1/2 * 1/3]
= 1/6 / [4/6]
= 1/4 = 0.25
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