I have $m$ white balls and $n$ black balls in an urn. I repeatedly draw balls from the urn without replacement until one color runs out. What is the probability that white will run out first?
I have the recurrence relation $p(m, n) = \frac{mp(m - 1, n) + np(m, n - 1)}{m + n}$ with $p(0, n) = 1$ and $p(m, 0) = 0$, and if you guess $p(m, n) = \frac{n}{m + n}$ you can show that it's a valid solution, but I have no clue how you'd derive it without the lucky guess or an intuition for why that's the solution.
Here's one way:
Consider pulling out the full sequence of all $n + m$ of them. White balls run out first if and only if the last ball in the sequence is black. The key observation now is that the probability of a given sequence happening forward is exactly the same as the probability of that sequence happening in reverse. Thus, the probability that the last ball is black is equal to the probability that the first ball is black; this probability is exactly $\frac{n}{m+n}$.