The Problem With Poker Odds

For those of you who may not have seen this, Sean Carroll in Cosmic Variance asked a poker question a little while ago.

[C]onsider the following three possible pairs of hole cards [in a game of Texas Hold’em]:

Jack-10 suited (e.g., a Jack of diamonds and a 10 of diamonds)

Ace-7 unsuited (e.g., an Ace of spades and a 7 of clubs)

Pair of sixes

The quiz is extremely simple, and should be easy for experts: assuming you don’t know what anyone else has, or yet what the board cards will be, which possibility is most likely to win at the end of the hand?

The answer:

Note that this is not really a poker-strategy question, it’s just a math question. There is a separate issue, which is “which is the best starting hand”, or for that matter “how should you play each hand?” — we’ll get to that later. But this is just a math problem — which is most likely to win if you choose to stay in the pot all the way to the showdown?

The answer, to nobody’s suprise, is: it depends! It does not depend on your position, or whether the betting is limit or no-limit — those might affect your strategy along the way, but at the end of the hand it’s just a matter of who has the best cards. What it does depend on is how many people you are playing against. The absolute probability that you will win obviously goes down if you are playing against more opponents with randomly-chosen cards, just because there are more ways they could beat you. But, much more interestingly, the ordering of which hand is best also changes.