Recursion: Basketball

Larry Bird and Michael Jordan are playing a one-on-one basketball match. Both start with 0 points. In each game of their match, Larry Bird has a 30 percent chance of winning, and outcomes are independent from one game to another. In each game, the winner is awarded 1 point and the loser is deducted 1 point. The first player to reach 2 points wins the match. What is the probability that Larry Bird wins the match?

Answer

This scenario can be treated similarly to a tennis deuce situation. Let $$p = 0.3$$ be Larry's chance of winning a single game.

- The probability that Larry wins two consecutive games is $$p^2 = 0.3^2 = 0.09$$ .

- The probability that the score resets (i.e., they split two games) is $2 \cdot p \cdot (1-p) = 2 \cdot 0.3 \cdot 0.7 = 0.42$ .

Using recursion:

P(\text{Larry wins match from start}) = \frac{0.09}{1 - 0.42} = \frac{0.09}{0.58} = \frac{9}{58}.

Thus, the probability that Larry Bird wins the match is $\displaystyle \frac{9}{58}$ .