- #1

musicgold

- 304

- 19

I am not sure I am solving this problem the correct way, as my answer seems too low.

**Problem**: In squash (a two player racquet game), the player who gets 9 points first wins the game. Whoever wins a rally gets the point, irrespective of who served the ball. I am ignoring other rules for the time being. If player A has the 60% chance of winning a rally, how many rallies, on average, will it take her to win a game?

**Simplified problem**: I assumed that it takes only 3 points, instead of 9, to win a game. Then I drew a decision tree and assumed that on average X number of rallies are required for player A to win a game. Based on that, I got the following four states, where W = won, L= lost.

W W W – Rallies required to win the game= 3

W W L - Rallies required = X + 1 - 2 = X -1

(subtracting 2, to account for 2 points already won, adding 1, to account for the wasted rally)

W L - Rallies required = X + 1 – 1 = X

L - Rallies required = X + 1

Then I got the following equation.

X = 3 * (0.6)^3 + (X-1) * 0.6^2 * 0.4 + X * 0.6 * 0.4 + (X+1) * 0.4

X = 0.784 X + 0.904

**X = 4.19 -The Average number of rallies to win 3 points for Player A.**

Is this a correct way of looking at this problem?

Thanks,

MG.