# Stopping moment regarding the ratio of black balls to all balls.

The urn contains at start one black ball, one white ball and one green ball. In the next steps, we randomly pick a ball from the urn and put it back with one additional ball coloured as the one we picked. Let $$t$$ be the step number when we first pick a black ball.

What is the probability that in the step number $$3t+2$$ the ball will be black?

I will only add that the first part of the exercise was to prove that the ratio of black balls to all balls after the n-th step is a martingale. I knew how to do the first part on the exam, but would like also to know how to do the harder part. Thanks in advance.