What is the probability of winning this game? 
A tennis player has a 60% chance of winning any given point in a tennis game. Calculate the probability that she will win the game within the first 6 points, stating any assumptions you make.
(A game is won when a player has won at least 4 points and won at least 2 more points than their opponent.)

So there are 3 choices:

*

*she wins 4 points with the probability 60%$^{4}$

*she wins 4 points and lose 1 point with probability 4 $\times$ 60%$^{4}$ $\times$ 40%

*she wins 4 points and lose 2 points 15 $\times$ 60%$^{4}$ $\times$ 40%$^{2}$
So the total probability of her winning the game is 0.69984. Is this right?

Using Stars and Bars, the correct answer is:
$$P(\text{win})=\binom{3}{0}0.6^4+\binom{4}{1}0.6^4\times0.4+\binom{5}{2}0.6^4\times0.4^2$$
$$=0.54432$$
She must win the last point, and so the sums represent 3 wins no losses, 3 wins 1 loss and 3 wins 2 losses in the preceding points of the game.
