In a contest, students a, B, and C will take turns to answer one randomly chosen question each time, starting with A. If a student gives a wrong answer to a question, he drops out of the game. The remaining two will continue to compete until one of them drops out. The last person remaining is the winner. Suppose that a student knows the answer to any one question independently of the other students, with probability p. Calculate the probabilities of the following cases: (a) C drops out first; (b) C drops out first, A next and B wins.

I would be grateful if you could please help me to solve this questions for me please

Thanks.

I think that the question is wrongly formulated, if I look at your other probability questions...

Anyway, I'll take it that this contest ends in a minimum number of rounds.

(a) P(C drops out first) = P(A knows answer) x P(B knows answer) x P(C doesn't know answer)
= p x p x (1-p)

Now, you can simplify.

(b) P(C drops out first, A next and B wins) = P(C drops out first) x P(A doesn't know the answer)

You already know the probability that C drops out first from part (a). You know the probability that A does not know the answer. And since only B is left, (s)he wins automatically.

Post what you get! :)