# Successive Wins

The three-set series containing two wins in a row are $$WWW$$, $$WWL$$, and $$LWW$$ (where $$W$$ is a win, and $$L$$ is a loss). Let $$p$$ be the probability that Elmer beats his father, and let $$q$$ be the probability that he beats the champion.

Using the champion-father-champion ordering, the probability of two wins in a row is:

$qpq + qp(1-q) + (1-q)pq = 2pq-q^2p$

Using the father-champion-father ordering, the probability of two wins in a row is: $pqp + pq(1-p) + (1-p)qp = 2pq - p^2q$

Since, $$p > q$$, we know that: $p > q$ $pq > q^2$ $p^2q > q^2p$

It follows that $$2pq - q^2p > 2pq - p^2q$$. Thus, Elmer should choose the champion-father-champion ordering.