Question

Kevin and Kira are in a history competition.

(i) In each round, every child still in the contest faces one question. A child is out as soon as he or she misses one question. The contest will last at least 7 rounds.

(ii) For each question, Kevin's probability and Kira's probability of answering that question correctly are each 0.8; their answers are independent.

Calculate the conditional probability that both Kevin and Kira are out by the start of round 7, given that at least one of them participates in round 3.

Answer 1

First, consider the probability that neither Kevin and Kira participate in round 3; that is, they fail before the third question is asked. Since their survival times are independent, the probability that Kevin gets to question 3 is 0.64, since he has to answer 2 questions correctly.

The probability that neither participates in round 3 is, therefore (1 - 0.64)^2 = 0.1296.

So the probability that at least one of them gets to round 3 is 1 - 0.1296 = 0.8704.

Now let's compute the probability that neither gets to question 7 and at least one of them gets to question 3.

This probability is equal to the probability that neither gets to round 7 minus the probability that neither gets to 3, which is

= (1 - 0.8^6)^2 - (1 - 0.8^2)^2 =0.544-0.1296

= 0.415

Therefore the desired conditional probability is simply 0.415/0.8704.

= 0.4768

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