Last Ball Drawn

A box contains $b \geq 1$ black and $w \geq 1$ white balls. Besides this box, there is a large pile of black balls (you may assume it's infinitely large). Each turn, $2$ balls are drawn uniformly at random from the box. If they are of the same color, a black ball from the pile is put into the box; otherwise, the white ball is put back into the box. This procedure is iterated until the box empties out and one ball is put back in. Find the probability that this last ball is white when $b = 20$ and $w = 10$.