I understand the process of solving this problem but there is one thing that I am confused about.
Why, in order to win, do we need $N(\tau)-N(s)=1$? I thought that the number of events in $(s, \tau]$ is given by $N(\tau)-N(s)$ so it would make sense that we want this to to be $0$, in order to win.
How can I get the probability of winning in this stopping game until nth event happens? Link to question asked before but this specific problem I am having was not addressed.