Tuesday, January 22, 2013

math problem: three of a kind

I was playing cards last night and an interesting situation came up.

In this game after the deal there are three community cards turned over in the middle of the card table. These three cards are called the flop. On this hand the three card flop happened to be three Kings.

The dealer remarked that there is a 1 in 256 chance that the flop is all the same rank - i.e. three of a kind. I thought about it for a little bit and told the dealer that this is wrong. The chance of the flop being three of a kind is not 1 in 256. I told him the correct answer. The dealer called the floor and they were confused and the floor walked away muttering saying he would look it up on the internet.

This isn't a difficult problem and I was able to work it out without writing anything down. A high school student with basic probability should be able to solve it. To simplify and standardize I will express it as follows.

A standard 52 card deck is shuffled so the cards are in random order. What is the probability that the top three cards in the deck are the same rank? It can be three of any kind Kings, Aces, 222, etc.

I'll let it sit for a couple of days and post the answer in the comments.

1 comment:

delsquared said...

turn over the top card. it can be anything. let's say it's a King.

turn over the second card. now for the second card there are 3 Kings of 51 remaining cards. so the chance the second card is a King is 3 in 51

turn over the third card. now there are 2 Kings left of 50 remaining cards. so the chance the third card is a King is 2 in 50.

probability works by multiplying together. so the chance of three of a kind is

1 * 3/51 * 2/50

= 1/17 * 1/25

= 1/425

so it will happen 1 in 425 times, or 424:1 odds against