Rogers gambles where his wealth gets increased by 3 dollars with the probability of p and decreases by 2

dollars with q of 1-p . He initially has 4 dollars and he will play till he has in total 5 dollars or he looses everything.

Derive a general expression in p and q terms for probabilities of all possible values his wealth can take when he stops playing.

For this game to be a martingale calculate p and also prove if optional stopping theorem will be valid for this case.