January 29, 2004
How to Count
We learn how to count today and applied that skill to figure out the probability distribution for the number of aces in a five card hand.
There are 4! = 24 ways to arrange four players. There are 2! =2 ways to arrange the two players on the left and 2! =2 ways to arrange the two on the right. Each team on the left appears 4 times so there are 24/4 = 6 possible teams formed from the two players on the left.
This is an instance of "the committee problem."
The general formula allows us to calculate that there are 2,598,960 possible five card hands that can be drawn from a standard 52 card deck.
The formula also allows us to calculate the number of hands that, for example, have 2 aces and 3 cards selected from the 48 that are not aces.
The probability of four aces in a five card hand is 48/2,598,960 or 1/54,145. If you drew a five card hand every morning, you would see a hand with four aces about once every 148 years.