Probability the seventh card dealt is a spade

Question

A standard 52-card deck is shuffled uniformly and dealt face down, one card at a time. What is the probability that the seventh card dealt is a spade? Does the answer differ for the first card?

Accepted Answer

By symmetry, a uniform shuffle makes each of the 52 cards equally likely to land in any fixed position. There are 13 spades, so any single position holds a spade with probability \frac{13}{52} = \frac{1}{4}. Position 7 is no different from position 1, or position 52: all give 1/4. Sanity check by conditioning on the first six cards. Whatever mix of spades appears in the first six, averaging over all equally likely arrangements returns the unconditional rate. The clean way to say it: the 13 spades are as likely to occupy any given seat as any other, so the marginal chance for seat 7 is 13/52. …

Probability the seventh card dealt is a spade

Your answer

More Probability questions