examday

/TMUA

Two lamps, $X$ and $Y$, are initially off. Each switches on once, at a time between $0$ and $6$ minutes. Their switching-on times $x$ and $y$ are chosen independently. The probability that $(x,y)$ lies in any part of the square $0\leq x\leq6$, $0\leq y\leq6$ is proportional to that part's area.

Lamp $X$ stays on for $2$ minutes, and lamp $Y$ stays on for $3$ minutes. Each then switches off.

Given that there is a time when both lamps are on, what is the probability that $X$ switches off before $Y$?