medium Games

Bestie Card

A release-checked medium problem for training Cards, Combinations and Permutations, Games.

Question

Rupert is playing a game where he first picks some integer $k \in \{1, 2, ..., 52 \}$. Given $k$, he then draws the first $k$ cards from a regular shuffled deck of $52$ cards. If the last card that Rupert draws is an ace, and if there exists exactly one ace remaining in the deck then Rupert wins. What $k$ should Rupert pick to maximize his probability of winning?

Practice focus

This Games problem is tagged Cards, Combinations and Permutations, Games. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.