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.


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