Best-of-Three Scoring

A player enters a best-of-three match (first to win two games wins the match). Each game is won with probability $p$ ($0 < p < 1$) and lost with probability $1 - p$, independently across games.

1. For $p = \dfrac{2}{3}$, find the probability that the player wins the match. 2. Two scoring schemes are proposed:

• Scheme 1: the match winner scores $3$ points, the loser scores $-2$;
• Scheme 2: the match winner scores $1$ point, the loser scores $0$.

Discuss which scheme gives the player the larger expected score, in terms of $p$.