Mathematics of Poker

Probabilities and Odds

Probability and odds are the most important mathematical concepts to learn when you're first starting poker. Indeed, they're used to figure out how likely you are to win a hand and whether or not it is profitable to call a bet from your opponent.

Outs

Outs are the count of the number of cards that if coming on the flop, turn, or river, would give you the winning hand.

For example, if you have 5♥4♥ and your opponent has A♠A♦ on a board of 3♥9♥J♠10♠ then you have 9 cards that improve your hand to the winning hand, a flush: A♥ K♥ Q♥ J♥ 10♥ 9♥ 8♥ 7♥ 6♥ 3♥ 2♥. These cards are your outs.

Similarly, if you are all-in preflop with A♦Q♣ against A♣K♦ then your outs to improve to the winning hand are the three queens (Q♦ Q♠ and Q♥).

Chance of Winning the Hand (Odds)

The chance of your hand winning is the number of cards that give you the best hand divided by the number of unknown cards remaining in the deck.

Winning chance = # of 'winning' cards / # of unknown cards in the deck * number of cards to come

Let's dive into an example!

We have 10♦10♠ and our opponent has K♦K♣ on a board of A♦J♣7♥4♠. This means we have two cards that give us the best hand (10♣ and 10♥), with only 1 card to come. We know 8 cards of the 52 in the deck meaning the number of unknown cards in the deck is 44. Therefore our equation will look like this:

Winning chance = 2 / 44 * 1 = 0.045 = 4.5% = 1:22

As you can see I've written out the answer in several different formats. These are all equal expressions so it's down to personal preference which one you find easiest to understand.

Pot Odds

There are two different calculations we're going to go through in this section.

We will first calculate our pot odds, then we will convert our pot odds into a percentage so we can see how much equity we need to profitably continue against our opponent's bet.

Pot odds = the size of the pot / the size of the bet you're facing

For example, the pot is $100, and your opponent bets $50. This means you have to call $50 into a $150 pot, which is where we see the expression "getting 3 to 1".

Pot odds = $150 / $50 = 3 to 1

Now we've got our pot odds, let's look at what we need to do to convert pot odds into a percentage. There are two ways we can do it, I will first give the full explanation of how we do it before showing you a way to make the math a little easier.

First, we need to figure out the size of the pot if we were to call the bet. In this example, the pot is $150 and it is $50 to call so our total pot will be $200 if we call. Then we divide the size of the call by the size of our total pot. In this case that will be $50/$200 which gives us 0.25.

Finally, to get our percentage we take the answer from our last sum (in this case, 0.25) and multiply it by 100 which gives us 25%. This is the amount of equity we need to profitably call the bet.

Now, instead of using the often large numbers of the pot, we can use our pot odds ratio that we figured out earlier to come to the same answer.

In our example, we will now add 1 to 3 to get our total pot size (which is 4) then divide 1 by 4 to get 0.25. From there we multiply by 100 (or just move the decimal point two places right) to get our percentage.

I imagine it will be much easier for most people to deal in smaller numbers rather than bigger numbers, especially in tournaments where chip denominations get very big!

Using Equity and Pot Odds In-game

These calculations are the easiest when we have complete information, but what about if we don't know exactly what our opponent has?

In this example, we'll see how can use pot odds and equity analysis together to see if a preflop call would be profitable:

• we have A♦K♦
• our opponent has gone all-in preflop for $100
• we've already put in $20 meaning we have to call $80 into $120 we've narrowed his range to AA, KK, QQ, and AK, based on how he's played

Now, we know from the section on pot odds that we divide the amount to call by the total pot if we were to call. In this case, $80/$200 which gives us 0.4. We then multiply 0.4 by 100 to get the percentage, in this case, 40%.

Now what we have to figure out if our hand has enough equity against our opponent's range to call. If we put our hand against that range into an equity calculator we see that A♦K♦ has 41.90% against that range, meaning we can profitably make the call.

Implied Odds

Implied odds as a concept is used to calculate the amount of money we expect to win if we hit our draw.

We may not be getting the right immediate pot odds, but if we think our opponent is going to pay us off in a big way then we may be getting the right implied odds to call and try and make our hand.

For example, if we have 7♦6♦ on A♣5♦8♠, it may be worth calling a large bet on the flop as if we make our straight it's likely we can win a big pot from any hand that our opponent has that is top pair or better.

This is the formula for working out implied odds:

((1 / EQ) * C) - (P + C)

Where C is the size of the current bet, P is the pot size after your opponent has bet and EQ is the likelihood of your hand improving.

This is quite a tricky formula to figure out so it's easier to input the variables into an instead.

Reverse Implied Odds

Reverse implied odds are, as the name implies, the opposite of implied odds. Whereas implied odds considers the money we could win if we improve to the winning hand, reverse implied odds considers the money we could lose if we improve to a second-best hand.

A good example of reverse implied odds is drawing to a straight on a board with a flush draw. Let's say we have 9♠8♠ on a board of 10♥J♥2♠ and we're facing a bet from our opponent. There are a couple of things to consider here.

Firstly, if a 7♥ or Q♥ comes we complete our straight but it brings a possible flush that could have potentially improve our opponent to a better hand than ours. Secondly, any Q gives AK a higher straight than ours which could lead to us losing a big pot.

This doesn't mean that you should never be drawing to a hand that's not the nuts, it just means you need to be aware of all the possible hands your opponent can have in any given spot, and be prepared to play cautiously if you see a lot of money going in on these dangerous turns and rivers.

Expected Value (EV)

The expected value is the average result of a given action if it were repeated over a significant number of samples (100s or even 1000s of times). Quite simply for any action we add up the win percentage and money won then subtract that from the lose percentage and money lost:

EV = (Win % * $ Won) - (Lose % * $ Lost)

If the result 'EV' is a positive one, that means that our action makes money or is +EV. However, if the result is a negative number that means our action loses money or is -EV. But how do we use this piece of poker math?

Let's look at an example hand:

• we have K♠K♦
• our opponent has gone all-in preflop for $100
• we need to decide whether or not to call the $100.

When we win we get the $100 from our opponent, when we lose we've lost the $100 we've called. We expect our K♠K♦ to win roughly 75% of the time against our opponent so let's plug these numbers into the equation to see if this is a +EV action.

EV = (0.75 * $100) - (0.25 * 100) = $50

As you can see the result was +$50 meaning that every time we make this call our expected value will be $50. If we were to run this hand only once we can see that the only two results are winning $100 or losing $100 but the more times we run this hand the closer our overall winnings come to $50 per hand.

Fold Equity

Fold equity is the equity gained based on the amount of our opponent's range we fold out when we bet. In many ways, it's the opposite of pot equity which is the equity we have against our opponent when they call.

Fold equity = probability of opponent folding * equity of opponent's folding hand

For example:

• we make a bet on the turn with 7♥6♥
• our opponent has K♠Q♦
• the board is A♥K♥4♠2♦
• we only have about a 20% chance of winning.

However, we think our opponent will fold about half the time if we bet a large size, so what's our fold equity?

Fold equity = 0.5 * 0.8 = 0.4 = 40%

We gain 40% equity by betting this hand and getting our opponent to fold half of the time.

Obviously, this kind of calculation is hard to make in-game when we don't know our opponent's hand. Yet, understanding the concept of fold equity will allow you to recognize situations like these where you can gain significant amounts of equity by betting and getting your opponent to fold a better hand.

Sklansky Dollars

"Sklansky dollars" is a concept created by the poker player and mathematics of poker whiz David Sklansky. He noticed, like many of us, that you can make the correct decision in poker but still end up losing money in the short term.

He wanted a way to put a value on his correct decisions even if they didn't return money every time so he came up with the Sklansky dollar.

To calculate Sklansky dollars:

• take your equity share of the pot at the time you're all in
• subtract your investment in the pot to see if you made 'Sklansky dollars'.

For example, you and your opponent are both all in preflop for $100. You have Q♠Q♦. Your opponent has J♦J♠. You're feeling good about life, thinking about how you're going to spend that extra $100, then shock horror! Your opponent spikes a J♥ on the river and you lose. You feel bad about losing $100 but you can comfort yourself by counting the Sklansky dollars you made instead.

Your equity share of the pot is $165 at the time you're all in, minus your investment in the pot of $100 means you've made 65 Sklansky dollars! This gives you a tangible way to see how profitable your decision was, regardless of the actual outcome of the hand.

Combinatorics Poker

Combinatorics as a poker math concept looks at counting the exact number of hand combinations that a player has in their range.

For example, we think our opponent has a range of AA, KK, QQ, and AK. If we were to count the exact number of combos in that range we will see there are 6 combos each of AA, KK, QQ, and 16 combos of AK for a total of 34 total hand combinations in our opponent's range.

However, if we hold A♠J♦ then this number changes as we have one of the aces, meaning there are fewer combinations of AA and AK that our opponent can have. The new numbers will be 3 combos of AA, 6 combos of KK and QQ, and 12 combos of AK for a total of 27 hand combinations.

As we can see just holding one ace reduces the number of AA hand combinations in our opponent's range by 50% and the number of AK combos by 25%. This is where the concept of 'blockers' comes from. By holding an ace, we 'block' our opponent from having all potential combinations of an AA hand range.

Another way combinatorics is used is in bluff-catching. For example, if there are 3 diamonds on the board, and we have 1 diamond in our hand, then by holding that diamond we reduce the number of flushes our opponent has in their range, thereby slightly weakening his range.

G-Bucks

GBucks, named after and created by elite PLO and NLHE player , takes the concepts of Sklansky dollars and combinatorics and builds on them to make it applicable to the concept of ranges as opposed to just a single hand. It's one of the most complex aspects of the mathematics of poker, but I will do my best to summarise.

Simply put, G-Bucks work as follows:

• take the amount of equity your hand has against each part of your opponent's range,
• multiply that equity by the number of combos of each hand,
• add the total of those results together,
• then divide by the total number of hand combos to get the average amount of equity your hand against your opponent's range.

I did say simply, right?

GBucks is a very advanced topic and could have a whole article dedicated to explaining it thoroughly, it's used by elite players in-game to calculate how their hand performs against their opponent's range.