The Mathematics of Poker, Basic Statistics, and Probability

Playing the Numbers Game

How to use math, statistics and probabilities to improve your game and increase your winning edge.

To start with, let me throw some numbers at you and see if you recognise what they are…..


  • 220-1 (0.45%)
  • 80/20
  • 11-1 (8.33%)
  • 95%
  • The answers will appear at the end of this article.


OK, If I said to you, ‘You’re a tight player on the button with pocket 5’s, and the grizzly, old guy you know nothing about is UTG and raises’ you can probably picture the scene quite well. It’s folded around to you and you either call, raise or fold. Being a tight player, you most likely fold and wait for a better spot. ‘Dan Harrington’s starting hands guide’ is your bible and you dare not argue with it.

But what have I actually told you here in terms of numbers? Apart from the two 5’s you hold, absolutely nothing which will help you make your decision! So let’s try again.

‘You’re a tight player on the button, playing a $1/2 cash game, with a full $200 buy-in in front of you, and you have pocket 5’s. The grizzly, old guy UTG you know nothing about has less than the min. buy-in of $80, let’s say $60, and raises 3xBB, so the pot stands at $9 when it reaches you – you need $6 to call’. What do you do?

Now we’re getting somewhere! For the sake of simplicity let’s say the blinds are tight, haven’t been very active lately anyway, and will likely fold unless they have a monster.

This is as much a poker math question as it is anything else. Now you can calculate the likelihood of winning money from this scenario – and that’s what poker is ultimately about. Understanding which scenarios which will make you money, and which will cost you money, and knowing the maths which will guide you to the correct answer.

What do you think the chances of your 55 being good is, if the old guy will only be raising UTG with 99+ or a big ace?  10%? 25%? 50%? If you don’t know the answer, then you need to learn it! You’re actually just under 40% against this range, which means you will usually need to improve your small pocket pair (or somehow outplay him in other ways).

What are the chances of improving your 5’s on the flop to the 2 pair or set you might need? 5%? 35%? Again, this is something which is easy to learn and memorise and will make your decisions much easier. The number is about 12% for hitting a set on the flop (closer to 20% if you include improving to 2 pair into the mix, but this might also give our opponent a bigger 2 pair so we can’t count most of that).

So, numerically speaking, we will not improve our hand very often on the flop. If we have called the old guys raise, what do we do when we miss the flop and he bets out? We basically lose some money! And what happens when we do hit our set or 2 pair on the flop? We trap the guy and double up! Yippee!! Err, no. Unfortunately, our short-stacked opponent doesn’t have anywhere near enough chips to do this.

It’s fairly simple math – we need our villain to have at least 70% of our stack to make this play with a small pair viable, a +EV situation. We need implied odds every time we play small pairs (or suited connectors, for example, whose numbers are similar) against a raise, and we can only get that when we are facing someone with a decent stack size. Winning a huge pot when your pairs improve makes up for all those times when they don’t and your chips head west.

This is actually a fairly simple, but important, example of how knowing the maths and probabilities in poker will improve your win-rate. Let’s try another….

You’re in the same game, same position, against the same opponent who now has a $200 stack like yourself, and your 6♦7 has reached the turn on a board of A♦89♠4♥. You figure your opponent for having paired an A on the flop as he has bet out on both the flop and turn.

The pot is currently $61 and you have a $20 bet to call. What would you do? Firstly, before getting into the math, let’s imagine what would happen if you call and a heart comes on the river. That would be a cooler; you wouldn’t know if he has a flush and or not – if he doesn’t and bets out he could be in big trouble. Let’s excuse that scenario from the argument as killing the action and do some counting.

How likely is it that you will fill your open-ended straight draw? 10%? 30%? The answer is that you have 8 outs on the river and so will hit it 17.2% of the time, but a quarter of those (the 5♥ and 10♥) can be discounted as above, so your actual figure is 13%. That’s giving you just under an 8-1 chance of winning, and you’re only getting odds of 3-1 on the pot. Barring some crazy and unlikely implied odds, the math tells you that it’s simply not a call.

Knowing these numbers are crucial to your game. A simple way to deal with situations like the last example, where you are behind, is to count your number of outs and double it to give you the winning probability on the river, or 4x it on the turn. Simpler still is to simply learn and memorise as many of these figures as you can!

Ok, now on to the answers from the introduction…

  • 220-1 (0.45%) Probability of being dealt pocket Aces
  • 80/20 Winning ratio of pair over pair
  • 11-1 (8.33%) Probability of making a Full House from 2 pair on the turn card
  • 95% Likelihood that you skipped to the end of the article to see if you were correct! If you did this, and didn’t know the other answers, you really need to go back and read the article!!!

About the author: Andrew Burnett is a chess and poker player, and a key contributor for both PokerVIP, and PokerTube. Follow his posts for real applicable knowledge and up to date news.

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