Probability of one golfer beating another with probabilities p(player 1 wins the hole), .... Two golfers are playing sudden death to decide a tournament. The first one wins a hole with probability p, the second one wins with probability q, and holes are tied with probability r. Holes are independent, and the game stops the first time someone wins a hole. What is the probability that the first player wins?

The probability of the first player winning is
$$p+rp+r^2p+\cdots=\frac{p}{1-r}=\frac{p}{p+q}\ .$$
Note that this agrees with the following intuitive argument: since draws are irrelevant, ignore $r$.  Then the ratio of probabilities is $p:q$, which gives the above probability for a first-player win.
