## Math, Poker, and Peter Winkler

Peter Winkler was kind enough to send me a copy of his new book Mathematical Mind-Benders. It is full of wonderful puzzles, at a level similar to the level aimed at by Martin Gardner (I have written about Peter’s work before): go and buy it! Here is a question (for which I will provide the answer in a day or so):

In poker, what is the best full house?

Comment: You may assume you are playing straight five-card stud poker (everyone gets five cards, all closed, no exchange] with (say) five companions, using a single normal deck. As a result of God owing you a favor, you are entitled to a full house, and you get to choose the full house you will get. Which should you choose?

This one I got (I get about 1 in 10 of the puzzles, mainly because either I have seen them or I know the relevant professional literature). I wonder how many professional poker players get it? Answer in a day or two. Comments with guesses welcome.

### { 3 } Comments

1. Michael Trick | September 7, 2007 at 3:43 pm | Permalink

The intended answer is any of AAA99, AAA88 down to AAA55. The reasoning is that a full house with AAA can only be beaten by 4 of a kind or a straight flush. The “5s” in AAA55 break up the A2345, 23456, 34567, 45678, and 56789 straight flushes, while choosing AAAKK, say, only breaks up the 9TJQK straight flush. Fewer hands that beat you equals a better flush in this argument.

As mathandpoker points out, however, the betting involved makes a difference. Someone sitting with KKKQQ is a lot more likely to give you lots of money than someone with garbage. So that points to AAA55 as perhaps the best choice. Better still would be for God to let you choose your opponents hands also!

I feel a little smarter having read the book.

2. Robyn | September 14, 2007 at 8:59 am | Permalink

Very interesting article. I was thinking AAAKK, but I wasn’t thinking about breaking up straights flushes because I was stuck on thinking about what the best possible full house was. I’ll definitely have to read this book now.

3. Steve | October 4, 2007 at 5:17 am | Permalink

Thought it would be AAAKK myself, which everyone seems to think would be the highest.

However the ‘best’ is a different story.

Good one for the pub that!

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1. Best Full House | September 6, 2007 at 7:55 pm | Permalink

[…] not sure what the answer to Michael Trick’s question about the best full house in 5 cards is.  I’m going to guess AAA22 unless there’s a […]