There are two boxes, one white, one black. The white box contains 3 white balls and 2 black balls. The black box contains four white balls and six black balls. This particular example 60%/40% and 40%/60% distribution split but in other versions of this problem the distribution could be uneven. I'm hoping for a generic algorithm/solution that can handle different distribution.
Balls are drawn from the boxes, the color is recorded, and the ball is returned to its box. You select a ball from the white box first. If the selected ball is white you choose from the white box next. If the ball is black you choose from the black box next. The probabilities of the first few selections are easy enough.
What is the probability that the 50th ball chosen is white? I'm thinking a very large decision tree would solve this but that seems an untenable manual process. Any suggestions much appreciated!