Aug 17, 2019 | Updated: 07:24 AM EDT

The complicated mathematics behind poker

Oct 30, 2018 03:14 AM EDT


What's the difference between an average poker player and a successful one? Some say that it's mainly down to luck and how the cards fall in any given situation. Others say it's down to strategy and their successful implementation as well as the ability to bluff with confidence.

What a lot of people fail to mention is that poker is a game based on mathematics, specifically probability and that is the basis of every successful strategy. Luck and strategy on their own are useless if you have little idea on the future possibilities of the flop, the turn or the river. Just look at what happened when man came up against machine.

In this article we'll show you how basic mathematics can improve your poker game.

What is probability?

The strand of mathematics that concerns poker is probability, which is the likely outcome of a certain event. In layman's terms probability can best be explained by the toss of a coin - when a coin is tossed, there are two possible outcomes, heads or tails.

Therefore the likelihood of the coin landing on one outcome is 1 in 2 or 50%. In Poker the probabilities are harder to calculate. In any given poker game the cards will be dealt from a deck of 52 cards with 4 different suits.

The probability of getting a King in is therefore 4 in 52 or 7.7% and then the chances of getting a second King is 3 in 51 as one card is already missing from the deck, leaving you with a 5.9% chance.

To work out your odds of receiving a pocket pair of Kings you have to multiply the probabilities of receiving each card so in this instance you would do the following sum;

(4/52) x (3/51) =

(12/2652) =

(1/221) =


That means that your chances of landing pocket Kings are incredibly low and that on average you should receive these cards once in 221 hands. Don't worry though, you don't need to get your calculator out and do these sums at the table, you simply need to know the basic probabilities of receiving certain hands.

What are pot odds?

In basic terms, pot odds are the ratio of the size of the pot in play to the cost of your next potential move. Pot odds are used by players to compare the chances of winning a hand with a future card, so they can estimate the value of the call.

Pot odds are usually expressed in ratios as they are easier to ascertain than percentages, but for ease and accessibility percentages are used by TV broadcasters.

How to work out your poker odds

Working out your pot odds requires a bit of practice, once you've got the basics sorted it will come as second nature to you at the poker table.

To work out your pot odds you'll firstly need to calculate your 'outs'. These are quite simply the cards that will help you improve your hand and make it better than your opponents.

Take this situation for example, you've been dealt the queen and nine of hearts, and the dealer lays out the ace of hearts, the king of hearts and the seven and four of spades. That means there are 9 hearts left in the deck, and you'll need just one of them to appear on the river for you to win.

From the cards that we can see, there are 46 cards remaining at play, meaning there are 37 cards that will see you lose and 9 that will see you win. In simple ratio terms that means you are 4 times as likely to lose as you are to win, leaving you with a 20% chance of success.

You might think that this seems quite complex and difficult to implement mid game, but it really isn't. The best thing you can do is enlist yourself in some soft or practice poker games and give it a go until you get your head around it.

Working out your pre-flop odds is a little more complex and will require a bit more skill, but again it's eminently achievable. Take a look at some of the key things to look out for below.

Premium hands: As discussed earlier in the article, the chances of getting pocket picture pairs are incredibly low. So it's best not to base your game on the chances of receiving these hands, but if you do pull them your chances of success will be considerably higher than your opponents.

The river flush: If you're just one card short of a full flush after the flop, your chances of drawing a full flop on the river are 34.97%, meaning you can be fairly confident of winning the hand.

Suits: Some players will tell you that playing any two cards because they're suited is a great tactic to employ. These players haven't worked out the odds, they're simply telling you about their anecdotal experiences. Playing two suited hands only improves your chances of winning by a measly 2.5%

The river odds: By the time the game reaches the river, your chances of making a pair increase by around 50%.

The better pair: On the occasion that two pairs go head to head, the higher pair wins roughly 80% of the time. So if you're holding queens, you might feel fairly confident, but be wary, if your opponent raises and re-raises, the likelihood is they're holding aces or kings.

Its race time: A coin-flip or race as some players call it, is simply a pair against two overcards because they each win about half of the time. If overcards are suited, the pair will win around 54% of the time, if they're not then it increases to 57% of the time.


Psychology, bluffing and strategy are the glamorous options when it comes to improving your poker game, but its basic probability mathematics that will make you a much better player. That's obviously disappointing to hear if you hate maths and it certainly isn't cool - you never see James Bond calculating probabilities at the table.

But if you choose to ignore the importance of probability in poker then you'll never fulfil your potential and win more money than you lose. So get familiarised with probability mathematics and put it into practice.

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