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Topic: How do you count the number of Full Houses (poker)
Replies: 19   Last Post: Jun 23, 2009 8:24 PM

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Posts: 12
From: West Paterson, NJ
Registered: 6/21/09
How do you count the number of Full Houses (poker)
Posted: Jun 21, 2009 8:06 PM
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Hi there, Math Geeks. I am a graduate math student (and aspiring math geek) who just finished taking undergraduate probability (I don't have any undergraduate math classes, so I have to go back and take all of them). This summer I am reading a book on Hold'em Poker which is showing the odds of making certain hands. I would like to figure out how they got these odds.

If this is posted in the wrong place, or is too difficult or easy for this forum, please help me find the right one.

The book says that poker hand rankings are based on how difficult it is to be dealt the hand in five cards. There are 2,598,960 different hands, which I get by doing (52 C 5), which means combinations of 5 cards chosen from 52, or 52!/(47! 5!) = 2,598,960.

Here are the odds the book publishes:

Royal Flush: 649,739 to 1
Straight Flush: 64,973 to 1
4 of a kind: 4,164 to 1
Full House: 693 to 1
Flush: 508 to 1
Straight: 254 to 1
3 of a kind: 46 to 1
Two pair: 20 to 1
One pair: 1.25 to 1
No pair: 1.002 to 1

I have figured out how they get the first two. There are four ways for a royal flush to be dealt, and 2,598,960/4 = 649,740 = 647,739 to 1.

A straight flush can be dealt 40 ways, 2,598,960/40 = 64,973 to 1.

I am trying to do the same for all the hands, but I cannot figure out how to count the number of ways the remaining hands can be counted. I thought that four-of-a-kind could be made 13 ways (right? Four Aces, or four Kings, ... or four Deuces), but when I divide 2,598,960/13, I do not get the book's answer. And how do you count the number of ways to make a full-house? It could be any cards as the two or the three, so how do you calculate that. And how do you calculate a straight, which requires specific cards?

So, the short version of my question is this: How do you count the number of ways to make the following hands:

4 of a kind
Full House
3 of a kind
Two pair
One pair
No pair

Here is some of what I've tried for other hands.

4 of a kind: 2,598,960/13 = 199,919 to 1 (wrong)
Flush: 2,598,960/(4(13 C 5)) = 503.8484 to 1 (wrong)
No pair: Sum up the ways of making all the other types of hands and subtract that number from 2,598,960.

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