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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. |
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Answer» <P>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 Learn More: In a Family of 3 children find probability of having at least one boy ... If you flip three fair coins, what is the probability that you'll GET all ... |
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