A little statistics puzzle…

Published on 2009-08-27 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.