## Recreational Math – How Much To Bet In Texas Hold ‘Em

In this post I will show how a little probability and a little elementary algebra can be used to determine how large of a bet you should make while playing No Limit Texas Hold ‘Em. (For a quick explanation of the rules, check out this video by Phil Gordon.)

### Set Up

You have a high pair, let’s say it is a pair of Aces (A♣, A♦). There are 400 chips in the pot, and you and only one other player remain in the hand.

The flop hits the table containing no Aces, but it does contain two hearts: 10♥, 6♦, 2♥. You know there is a decent chance that your opponent holds two hearts in his hand, because he likes to play suited cards.

The question is – How much should you bet? Paraphrasing the “Fundamental Theorem of Poker”, you should bet enough so that your opponent is making a mistake if he calls.

### What Are The Odds Against Your Opponent Completing Their Flush?

From your opponent’s point of view, he has seen 5 cards and 4 of them are hearts. That means that there are still 9 hearts in the deck that would complete the flush, and the other 38 would not. So the odds against the flush are 38:9, or approximately 4.2:1.

### How Much Should You Bet?

If the opponent is being offered less than 4.2 to 1 on his wager, he is making a mistake to call. The goal is to make a bet (x) large enough so the ratio of the pot (400 + x) to the amount he would have to add (x) is less than 4.2. Here comes the algebra. We will determine the value of x for which (400 + x)/x = 4.2, this will give us the mathematically fair wager.

(400 + x)/x = 4.2

400 + x = 4.2x      (Note that x cannot equal 0 in this problem.)

400 = 3.2x

x = 400/3.2 = 125

A bet of \$125 creates a pot of \$525, offering your opponent 525:125 (or 4.2:1) odds for calling.

Any wager greater than \$125, will offer your opponents insufficient odds to call. For example, a bet of \$200 increases the pot to \$600 and offers your opponent only 3:1 odds.

The trick is to size your bet as large as you can while still getting your opponent to call.

### And If There Is No Heart On The Flop …?

You get another chance to force a bad bet. Suppose you bet \$200 and your opponent called, the pot is now \$800. The odds against completing the flush on the next card are now 37:9 or approximately 4.1:1. The same strategy still applies. In general, with a pot size of P, the bet size x should satisfy the following inequality:

(P + x)/x < 37/9

We find that if x > 9 P / 28, that our opponent will make a mistake to call the bet. If the pot is \$800, this equates to a bet larger than\$257.14. Again, the important idea is to make as large of a bet as you can that you feel will still be called.

-George

I am a mathematics instructor at College of the Sequoias in Visalia, CA. Each Friday my blog contains an article on recreational mathematics. Let me know if there are other topics you’d like me to address. You can reach me hrough the contact page on my website – http://georgewoodbury.com.

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• 1. igor  |  May 14, 2010 at 9:24 am

thx for the post.cant get enough info when it comes to texas holdem poker!

• 2. jobrist  |  March 2, 2012 at 7:07 pm

I don’t agree with the odds you first presented after the flop. You calculated only the odds on the next card. You should have calculated the odds assuming two upcoming cards.

The odds of not getting the heart on the first card is 38/47, and not getting the heart on the second card is 37/46, making the odd of no hearts for both cards (38*37)/(47*46) = 0.65. This makes the odds of getting a heart in the first OR second card (1 – 0.65) = 0.35. The lose:win odds are there for 0.65:0.35 = 1.68:1, not 4.2:1, of course, this is still assuming the other guy won’t win with a flush as well.