Head Start in an Exponential Race

Let $X \sim \operatorname{Exp}(\lambda)$ and $Y \sim \operatorname{Exp}(\mu)$ be independent. Player A finishes at time $X$ and player B finishes at time $Y + c$ where $c > 0$ is a head start for player A (player B starts $c$ time units later).

(a) Derive $P(X < Y + c)$ — the probability that A finishes before B.

(b) Show that as $c \to 0$, the result recovers the standard competing-exponentials formula.

(c) Evaluate for $\lambda = 3$, $\mu = 2$, $c = 1$ and interpret how the head start affects A's winning probability.