Problem #PRU-65298

Problem

In the final of a contest of performances on March 8 two performances were left. In the first, $n$ pupils from the class 5A performed, and in the second – $n$ pupils of class 5B. At the play there were $2n$ mothers of all $2n$ students. The best performance is chosen by a vote by the mums. It is known that every mother, with a probability of $\frac{1}{2}$, votes for the best performance and with a probability of $\frac{1}{2}$ – for the performance in which her child participates.

a) Find the probability that the best performance will win with a majority of votes.

b) The same question, if more than two classes have reached the finals.