Ballot problem of probability- probability of the last vote  written by Sheldon M Ross. Here conditioning has been used on the last vote. Pn,m = P{A always ahead|A receives last vote} * P{A receives last vote} + P{A always ahead|B receives last vote} * P{B receives last vote} But how can P{A receives last vote} be equal to n/(n+m) ? The last vote can go to only one of all the candidates, so it should be 1/number_of_candidates (Though the number of candidates is not mentioned here) The same applies for B as well. Can anyone please explain this?

The last vote cast must go to either $A$ or $B$. The question only cares about the two relevant candidates, so we can assume that there are a total of $m+n$ votes cast, $n$ of which went to $A$ and $m$ of which went to $B$. Thus, if we only look at the last vote, since each ordering is equally likely, the probability that it is one of these $n$ that went to $A$ is $\frac{n}{n+m}$.