Here is the problem setup:

Two people play a game of throwing balls at a pin. Sally hits the pin 70% of the times she throws, and Bill hits the pin 40% of the times he throws. The game is won by the person who hits the target first. If Bill throws first each time, find the probability he will win.

With simulation, I get Bill's probability of winning to be around 0.52 (rounding to two decimal places). **How to solve this problem analytically?**