An upcoming tennis player can win a prize if she wins two consecutive matches in a three-game series against the #1 and #2 players alternately. She can choose either of the matchups: 121 or 212. Which scheme has a better chance of winning the prize?
Let p1 be her chance to defeat #1 and p2 to beat #2; p1 < p2 (#1 is a better player than #2).
Matchup 121
The probability of winning two consecutive matches in the 121 scheme is: p1 x p2 x p1 + p1 x p2 x (1-p1) + (1-p1) x p2 x p1 = p1p2(2 – p1)
Matchup 212
The probability of winning two consecutive matches in the 212 scheme is: p2 x p1 x p2 + p2 x p1 x (1-p2) + (1-p2) x p1 x p2 = p1p2(2 – p2)
Therefore, it reduces to a comparison bewteen p1p2(2 – p1) vs p1p2(2 – p2) or (2-p1) vs (2-p2).
Since p1 < p2, (2-p1) > (2-p2). So she must go for matchup 121.
Reference
Fifty Challenging Problems In Probability: Frederick Mosteller
