HMMT-Nov 2016 guts, Problem 10

Michael is playing basketball. He makes $10 \%$ of his shots, and gets the ball back after $90 \%$ of his missed shots. If he does not get the ball back he stops playing. What is the probability that Michael eventually makes a shot?

Solution: We find the probability Michael never makes a shot. We do casework on the number of shots Michael takes. He takes only one shot with probability $\frac{9}{10} \cdot \frac{1}{10}$ (he misses with probability $\frac{9}{10}$ and does not get the ball back with probability $\frac{1}{10}$ ). Similarly, he takes two shots with probability $\frac{9}{10} \cdot \frac{9}{10} \cdot \frac{9}{10} \cdot \frac{1}{10}$, three shots with probability $\left(\frac{9}{10}\right)^{5} \frac{1}{10}$, and so on. So we want to $\operatorname{sum} \sum_{i=1}^{\infty}\left(\frac{9}{10}\right)^{2 i-1} \cdot \frac{1}{10}=\frac{\frac{1}{10} \cdot \frac{9}{10}}{1-\frac{81}{100}}=\frac{9}{19}$. Then the probability Michael makes a shot is $1-\frac{9}{19}=\frac{10}{19}$. $\fbox{10 / 19}$.

Answer: 10 / 19