Learn / Library / Probability

Winning from deuce

ProbabilityHard

A tennis-style game is tied at deuce. The game continues point by point, and a player wins once they are ahead by $2$ points. Anna wins each point with probability $0.6$ , independently of other points. What is the probability that Anna wins the game? Give a fraction a/b in lowest terms.

Checking an answer needs JavaScript. The hint and the solution open without it.

Hint

Look at the next $2$ points: either someone wins the game or the score returns to deuce.

Solution

Anna wins the next $2$ points with probability $0. 6^{2} = 0.36$ .
Anna loses the next $2$ points with probability $0. 4^{2} = 0.16$ .
Otherwise, with probability $1 - 0.36 - 0.16 = 0.48$ , the score is back at deuce, so the situation repeats.
Hence $P = 0.36 + 0.48 \times P$ , which gives $P = \frac{0.36}{0.52}$ .
This simplifies to $\frac{36}{52} = \frac{9}{13}$ .
The answer is $\frac{9}{13}$ .

Answer: $\frac{9}{13}$

Next question Back to the library