In an election, candidate $A$ receives $n$ votes and candidate $B$ receives $m$ votes, where $n>m$. Assume that in the count of the votes all possible orderings of the $n+m$ votes are equally likely. Let $P_{n, m}$ denote the probability that from the first vote on $A$ is always in the lead. Find
(a) $P_{2,1}$
(b) $P_{3,1}$
(c) $P_{n, 1}$
(d) $P_{3,2}$
(e) $P_{4,2}$
(f) $P_{n, 2}$
(g) $P_{4,3}$
(h) $P_{5,3}$
(i) $P_{5,4}$
(j) Make a conjecture as to the value of $P_{n, m}$.