I'm trying to get my head around this problem, and I think I have a way to think about it.
So let's say I have $a$ white balls and $b$ black balls in a bag initially. I take out a ball and if it's white, I put it back and if it's black, I replace that black ball with a white ball. Let $M_n$ be the expectation of the number of white balls in the bag after $n$ moves.
I want to show
$M_{n+1} = \left(1-\frac1{a+b}\right)M_n + 1$
I think a way to do this problem is by writing something like this
Let $w_n =$ number of white balls after $n$ goes.
Let $b_n =$ number of black balls after $n$ goes.
$M_{n+1} = P(\text{Pick a WhiteBall})(w_n) + P(\text{Pick a BlackBall})(b_n)$
But I'm not sure where to go from here. In other words, how do I go from probabilities to expectations in a sequence problem such as this?