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The Paradox of the Second Ace

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You’re watching four statisticians play bridge. After a hand is dealt, you choose a player and ask, “Do you have at least one ace?” If she answers yes, the chance that she’s holding more than one ace is 5359/14498, which is less than 37 percent.

On a later hand, you choose a player and ask, “Do you have the ace of spades?” Strangely, if she says yes now the chance that she has more than one ace is 11686/20825, which is more than 56 percent.

Why does specifying the suit of her ace improve the odds that she’s holding more than one ace? Because, though a smaller number of potential hands contain that particular ace, a greater proportion of those hands contain a second ace. It’s counterintuitive, but it’s true.