Probability the second card dealt is an ace

Question

A standard 52-card deck is shuffled uniformly and dealt face down, one card at a time. What is the probability that the second card dealt is an ace? What about the 17th card?

Accepted Answer

By symmetry, a uniform shuffle makes every card equally likely to end up in any fixed position. Position 2 is just as likely to hold any particular card as position 1 is, so P(	ext{2nd card is an ace}) = \frac{4}{52} = \frac{1}{13}. The same holds for the 17th card, or the 52nd: any fixed position holds an ace with probability 1/13. Sanity check by conditioning. Condition on the first card: P(	ext{2nd is ace}) = \frac{4}{52}\cdot\frac{3}{51} + \frac{48}{52}\cdot\frac{4}{51} = \frac{12 + 192}{52 \cdot 51} = \frac{204}{2652} = \frac{1}{13}. \checkmark The conditioning is only needed if you se…

Probability the second card dealt is an ace

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