The Thirteenth Card

Question

A standard 52-card deck is shuffled. What is the probability that the 13th card from the top is an ace?

Accepted Answer

Exactly frac452 = frac113, the same as for the top card. A uniform shuffle is exchangeable: every one of the 52 cards is equally likely to occupy position 13, so the marginal distribution of the card in any fixed position is uniform over the deck. Four of the 52 cards are aces: P(text13th card is an ace) = frac452 = frac113. Approach 2, the long way, telescoped You could condition on how many aces appear in the first 12 cards and sum over cases. Every term of that sum is the expectation of a hypergeometric ratio, and the sum collapses to frac113, symmetry guarantees the collapse before you compute a single term, which is the entire point. The principle Unconditional marginals don't depend on position under a symmetric shuffle. The information that would change the answer, what the first 12 cards actually were, was never given. Distinguishing "no information revealed" from "information revealed and ignored" is the cleanest habit this question builds; it is the same reason the second card is red with probability frac12, and the same reason a randomly cut deck has an ace on top with probability frac113.

The Thirteenth Card

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