On Gambler's ruin problem in some models

2-player probability games is an important subject of probability theory. In this thesis, we considered four different two-player probability games. We calculated the expected value of stopping times of the games. Since the rules of the games are complicated and it is not possible to calculate all possible shuffles, we calculated the results using the Monte Carlo method on a computer. We showed that the expected value of duration of the games are of the order square of number of cards as we expected.