The card after the first ace

Question

A standard 52-card deck is well shuffled. Cards are dealt face up one at a time until the first ace appears. Look at the next card.

Is it more likely to be the ace of spades, or the two of clubs? Give the exact probability of each.

Most people feel that an ace is "used up" and the two of clubs is more likely. Check that feeling.

Accepted Answer

They are equally likely: each has probability exactly 1/52. The insertion argument. Take the two of clubs out of the deck and shuffle the remaining 51 cards. Now insert the two of clubs into one of the 52 possible positions (before the first card, between any two cards, or after the last). If each position is equally likely, the result is a uniformly shuffled 52-card deck. One of the 52 positions is "immediately after the first ace of the 51-card deck". So the probability that the two of clubs sits immediately after the first ace is exactly 1/52. Now do the same with the ace of spades. Remove…

The card after the first ace

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