Judges running the popular TV show MasterChef have been taking bribes from contestants. In 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 he/she will be allowed to stay with a probability of 1 . If the contestant has not been bribing the judges, he/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 each of the two-episode. That means, if a contestant bribes them in the first episode, he/she will bribe them in the second episode 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 he/she was bribing the judges? b) If you pick a random contestant, what is the probability that he/she is allowed to stay during both of the first two episodes? c) If you pick a random contestant who was allowed to stat during the first episode, what is the probability that he/she gets kicked off during the second episode.
Answers
Given : the contestant has been bribing the judges he/she will be allowed to stay with a probability of 1 . If the contestant has not been bribing the judges, he/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 each of the two-episode. That means, if a contestant bribes them in the first episode, he/she will bribe them in the second episode too (and vice versa).
To Find : a) If you pick a random contestant who was allowed to stay during the first episode, what is the probability that he/she was bribing the judges?
b) If you pick a random contestant, what is the probability that he/she is allowed to stay during both of the first two episodes?
c) If you pick a random contestant who was allowed to stat during the first episode, what is the probability that he/she gets kicked off during the second episode.
Solution:
Total contestants = 4x
then Bribing = x
not bribing = 3x
P Staying = x/4x + (1/3) (3x/4x) = 1/4 + 1/4 = 1/2
probability staying bribing the judges = 1/4
probability that he/she was bribing the judges = (1/4)/(1/2) = 1/2
If you pick a random contestant, what is the probability that he/she is allowed to stay during both of the first two episodes
= 1/4 chances bribing and 3/4 chances not bribing
(1/4) * 1 * 1 + ( 3/4)(1/3)(1/3)
= 1/4 + 1/12
= 4/12
= 1/3
If you pick a random contestant who was allowed to stay during the first episode, what is the probability that he/she gets kicked off during the second episode.
Allowed to stay in one = 1/2
Kick off = (3/4) (1/3) ( 2/3) = 1/6
P = (1/6)/(1/2) = 1/3
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