Alice and Bill fight a paint-ball duel. Nobody has been hit so far and they are both left with one shot. Being exhausted, they need to take a breath before firing their last shot. This takes $A$ seconds for Alice and $B$ seconds for Bill. Assume these times are exponential random variables with means $1 / \alpha$ and $1 / \beta$, respectively.

Find the distribution of the (random) time that passes by before the next shot is fired. What is its standard deviation? What is the probability that Alice fires the next shot?

Assume Alice has probability $1 / 2$ of hitting whenever she fires whereas Bill never misses his target. If the next shot is a hit, what is the probability that it was fired by Alice?