An interesting puzzle I came across:
For $T>1$, observe a Poisson process of rate $1$ on the time interval $(0,T)$. Every time we observe a point, we may choose to stop. To win the game, we must stop on the last point before time $T$. Else, if we stop at $t$ and there is another point in $(t,T)$, we lose. Also, if we never stop, we also lose. What is the best strategy for playing this game?
My hypothesis is that we should stop as close to $\lfloor T\rfloor$ as possible each time to maximise our chances of winning, but a proof is more difficult to come up with. Can you prove my claim correct?