Suppose that my children, Aidan and Julia, agree to play a game where they will both toss two fair dice. Let's define our random variables. Let X_An represent the number of even numbers tossed by Aidan in round n and let X_Jn represent the number of even numbers tossed by Julia in round n. Now we define Y_n = X_Jn - X_An. Of course, yn represents the observed value of Y_n. If y_n < 0, then Julia will pay Aidan $|y_n|. If y_n > 0, then Aidan will pay $yn to Julia. If y_n = 0, then no money will change hands. Also, if either player does not have enough money left to pay what they owe, then they will pay what they have left. Aidan and Julia will each start with $10. The game will end when one of them has all of the money.

Define states {0, 1, 2, 3, ....., 19, 20} to be the amount of money held by Julia. If P is the transition matrix for the Markov chain for this game, then what is P_{10, 11}?

A) 1/16 B) 1/8 C) 1/4 D) 3/8