Yet Another Unfair Game

You were 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,​ you lose one dollar.

You knew it is an unfair game, but you wanted to try your luck, so you decided to play with two dollars. You ended up losing your two dollars, but the game owner decided to lend you as much money as you want to spend in the game. You decided to play until you recover your two dollars, and without owing anything to the owner.

Let \(P\) be the probability that you achieve your goal, find \(\left\lfloor {10^{10} e^P + 0.5} \right\rfloor \).


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