6. Probabilistic Voting (6-9). In the final of a play competition for March 8, two plays made it to the finals. In the first play, $n$ students from 5th grade A participated, and in the second play, $n$ students from 5th grade B participated. At the play, $2n$ mothers of all $2n$ students were present. The best play is chosen by the mothers' votes. It is known that each mother votes for the best play with a probability of $\frac{1}{2}$ and for the play in which her child participates with a probability of $\frac{1}{2}$.

a) (1 pt.) Find the probability that the best play will win by a majority of votes.

b) (1 pt.) The same question, if more than two classes made it to the final.
# Solution.

a) Let's call a mother confident if her child is playing in the best play. A confident mother will vote for the best play with a probability of 1. There are exactly $n$ confident mothers, so the best play will receive at least half of the votes - it will get, at the very least, the votes of all confident mothers. The only case when the best play does not win by a majority of votes is a tie, when the worst play also receives votes from all uncertain mothers. This is only possible if all uncertain mothers vote for their own children. The probability of this event is
$p=\left(\frac{1}{2}\right)^{n} \cdot$ Therefore, the probability that more votes will be cast for the best play is $1-\left(\frac{1}{2}\right)^{n}$.

b) Suppose $s>2$ classes made it to the final, with each class having the same number $n$ of participants in their play. The best play will receive at least $n$ votes - one vote from each confident mother. Each of the other plays will receive no more than $n$ votes from all mothers. If at least one uncertain mother votes for the best play, it wins. Therefore, the best play will not win by a majority of votes only if all uncertain mothers (there are exactly $(s-1) n$ of them) vote for their own children. The probability of this event is $\frac{1}{2^{\text {sn-n}}}$. Therefore, the probability that the best play will be recognized as the best is $1-\frac{1}{2^{n s-n}}$.