Two players $A, B$ of equal skill are playing a set of games ; $A$ wants $2$ games to complete the set, and $B$ wants $3$ games: compare their chances of winning.

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Solution

In order to win, A must win 2 games before B wins3. Thus
A's chance $= {(\frac{1}{2})}^{2} {1 + 2. \frac{1}{2} + \frac{2.3}{1.2} {(\frac{1}{2})}^{2}} = \frac{11}{16}$
So B's chance $= 1 - \frac{11}{16} = \frac{5}{16}$
$∵$ their win events are equally likely
$∴ \frac{A}{B} = \frac{11}{15}$

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Two players A,B of equal skill are playing a set of games ; A wants 2 games to complete the set, and B wants 3 games: compare their chances of winning.

In order to win, A must win 2 games before B wins3. ThusA's chance=(12)2{1+2.12+2.31.2(12)2}=1116So B's chance=1−1116=516∵ their win events are equally likely∴AB=1115

Two players $A, B$ of equal skill are playing a set of games ; $A$ wants $2$ games to complete the set, and $B$ wants $3$ games: compare their chances of winning.

In order to win, A must win 2 games before B wins3. Thus
A's chance $= {(\frac{1}{2})}^{2} {1 + 2. \frac{1}{2} + \frac{2.3}{1.2} {(\frac{1}{2})}^{2}} = \frac{11}{16}$
So B's chance $= 1 - \frac{11}{16} = \frac{5}{16}$
$∵$ their win events are equally likely
$∴ \frac{A}{B} = \frac{11}{15}$