A little statistics puzzle…

I hand you a deck of cards, which you shuffle, and deal me a random 5 card poker hand. I announce that I have an ace in my hand. What are the odds that I have a second ace in my hand?

Now, play the game, and deal me another hand. I announce that I have the ace of spades in my hand. Now, what are the odds that I have a second ace in my hand?

Explain.

Addendum: This is known as the “paradox of the second ace”. It is usually expressed as a bridge problem, with 13 card hands, but I misremembered it, so here it is a poker problem. Solve the bridge problem if you like.

Addendum2: The paradox is easier to understand if you limit the size of hands (say to two cards) and the number of suits.

One thought on “A little statistics puzzle…

  1. Mark VandeWettering Post author

    counting hands with at least one ace
    886656
    counting hands with more than one ace
    108336
    counting hands with the ace of spades
    249900
    counting hands with the ace of spades, and at least one more ace
    55320

    So the odds of having two aces is only 108336/886656 if you have one ace, but rises to 55320/249900 if you have the ace of spades.

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