The critical point to recognize is that we need to look at the exact same event from different perspectives. Conceptually, there appears to be a single event, which is described as “the starting player wins with initially \(m\) white, and \(n\) black balls”. The probability of this event can be designated as the function:

\[p(m,n)\]

Let’s think concretely here. Adam and Bob are about to play the game. Adam will go first. So we are interested in “the probability that Adam wins with \((m,n)\) balls”, which coincides with “the probability that the starting player wins with \((m,n)\) balls”. How can Adam win?

  1. either he draws a white ball in his first attempt;
  2. or, he draws a black ball in his first attempt and Bob does NOT win the game that he starts with \((m, n-1)\) balls.

Then, the probability that Adam wins, namely \(p(m,n)\) has to be the sum of the probabilities of these two events:

\[p(m,n) = \frac{m}{m+n} + \frac{n}{m+n}(1 - p(m, n-1))\]

The game can last maximally until no black balls are left in the jar, since \(p(m,0)=1\) for any \(m>0\).