2.II.11F

A normal deck of playing cards contains 52 cards, four each with face values in the set $\mathcal{F}=\{A, 2,3,4,5,6,7,8,9,10, J, Q, K\}$. Suppose the deck is well shuffled so that each arrangement is equally likely. Write down the probability that the top and bottom cards have the same face value.

Consider the following algorithm for shuffling:

S1: Permute the deck randomly so that each arrangement is equally likely.

S2: If the top and bottom cards do not have the same face value, toss a biased coin that comes up heads with probability $p$ and go back to step $\mathrm{S} 1$ if head turns up. Otherwise stop.

All coin tosses and all permutations are assumed to be independent. When the algorithm stops, let $X$ and $Y$ denote the respective face values of the top and bottom cards and compute the probability that $X=Y$. Write down the probability that $X=x$ for some $x \in \mathcal{F}$ and the probability that $Y=y$ for some $y \in \mathcal{F}$. What value of $p$ will make $X$ and $Y$ independent random variables? Justify your answer.