Probability the bottom card is the ace of spades

Question

A standard 52-card deck is shuffled uniformly and placed face down. What is the probability that the bottom (52nd) card is the ace of spades? And that it is any ace?

Accepted Answer

By symmetry, each of the 52 cards is equally likely to occupy any fixed position, including the bottom. There is exactly one ace of spades, so P(	ext{bottom card is the ace of spades}) = \frac{1}{52}. For any ace there are 4 favorable cards among 52 positions: P(	ext{bottom card is an ace}) = \frac{4}{52} = \frac{1}{13}. Neither answer depends on the bottom being "last." The very first card, or the middle card, has the same 1/52 chance of being the ace of spades and the same 1/13 chance of being an ace. Takeaway. A fixed position in a shuffled deck is a uniform draw over all 52 cards, so "s…

Probability the bottom card is the ace of spades

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