# Backgammon match

• Mar 10th 2010, 11:40 AM
essedra
Backgammon match
I don't know how to solve this probability question, at first I thought it was a Bernoulli Random Variable question, but it's not. I mean, this is more like counting... but I don't know how to solve it... Can anyone help me? Here's my question:

Consider a backgammon match with 17 games, each of which can have one of two outcomes: win (1 point), or loss (0 points). Find the number of all possible distinct score sequences under the following alternative assumptions.
All 17 games are played.
The match is stopped when one player wins more than half the games.
• Mar 10th 2010, 11:50 AM
icemanfan
If all 17 games are played, all possible score sequences are determined by which games of the 17 player 1 wins.

So either player 1 wins the first game or he loses it, and the same goes for the other 16 games. Thus, there are two scoring possibilities for each game. Since the result of any one game is independent of the others, we can multiply the number of possibilities for each game together, and determine that there are 2^17 score sequences.

The other problem is a bit trickier, because we have to stop counting when one player reaches 9. Do you have any idea how to count the possibilities here?
• Mar 10th 2010, 03:47 PM
Plato
Quote:

Originally Posted by essedra
Consider a backgammon match with 17 games, each of which can have one of two outcomes: win (1 point), or loss (0 points). Find the number of all possible distinct score sequences under the following alternative assumption: The match is stopped when one player wins more than half the games.

Suppose that player A wins.
That can be done in $\sum\limits_{k = 0}^8 {\binom{8+k}{k}}$ ways.