A little statistics puzzle…

August 27, 2009 | General | By: Mark VandeWettering

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.

Comments

Comment from Mark VandeWettering
Time 8/28/2009 at 10:09 am

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.