- #1
musicgold
- 304
- 19
Hi,
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.
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.