An Unfair Game

You are invited to play a game. The game is played with a 6-sided fair dice. You choose two different numbers from the dice, and then the dice is rolled. If the result is one of the numbers you chose, you win a dollar, otherwise,‚Äč you lose one dollar. You know it is an unfair game, but you want to try your luck, so you decided to play with two dollars until you get three dollars or until you lose the two dollars you already have. Let \(\frac{a}{b}\) be the probability that you end the game with three dollars, where \(a\) and \(b\) are coprime positive integers. Find \(a^b\).

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