An Intro to Probability in Poker
Gabriel Bridger · 5 October 2026
You can use probability to play poker?
Intro
The aim of this post is to introduce the idea of using maths to play poker, rather than simply thinking 'oh I like my cards, I'm going to play'. This is because there are a lot of parallels between poker and trading, whether that be position sizing, risk management or just staying level-headed when things go wrong.
When looking at poker theory from this angle, everything is seen through the law of large numbers. This means that if you used these techniques consistently, in the long run they would work in your favour on average, but that doesn't mean there isn't variance. You can think about it like the S&P 500: it has averaged roughly 10% growth a year, but naturally some years it does less than 10% and some years it does more.
This post assumes you know the rules of Texas Hold 'em and the hand rankings (what beats what). Everything beyond that is explained as we go.
Basic terminology
Stack
Your stack is just the amount of chips you currently have. For the rest of the post, and whenever talking about any sort of poker strategy, you refer to your stack in terms of how many big blinds (bb) you have. So for example if you have 10,000 chips and the blinds are 25/50, your stack is $\frac{10000}{50} = 200$ big blinds.
We do this because it means you're always thinking about your stack in relative terms. With 10,000 chips and blinds of 25/50 you have 200bb, but if the blinds were 100/250 you'd only have 40bb, meaning you have much less effective money to play with even though in absolute terms the stack size is the same.
Table position
Where you're sat relative to the dealer button changes which hands you should play before the flop, and how easy they are to play after it. At a 6-player table the positions are as follows:
| Position |
Name |
Seats from the button |
When they act |
| BTN |
Button (dealer) |
0 |
Last to act after the flop, the best seat |
| SB |
Small Blind |
1 |
Posts half a blind; first to act after the flop |
| BB |
Big Blind |
2 |
Posts a full blind; last to act before the flop |
| UTG |
Under the Gun |
3 |
First to act before the flop |
| HJ |
Hijack |
4 |
Second to act before the flop |
| CO |
Cutoff |
5 |
Just before the button |
This then lets us describe when someone is 'out of position', which is when after the flop you have to act before a particular opponent in a given hand. The button is never out of position after the flop, while the blinds almost always are.
Hand notation
- Your hole cards are the two cards dealt to you, and the board is the community cards in the middle.
- AK means an ace and a king of any suits, and QQ means a pair of queens.
- A pocket pair is when both your hole cards are the same rank, like 7♣ 7♦.
- A set and trips both mean three of a kind. A set is a pocket pair plus a matching card on the board (7♣ 7♦ on a 7♠ K♥ 2♦ flop), while trips is one card in your hand plus a pair on the board (A♠ 7♣ on a 7♠ 7♥ 2♦ flop).
- A street is each round of betting: pre-flop, the flop, the turn and the river.
- A draw is a hand that needs one more card to become strong:
- Flush draw: four cards of the same suit, needing a fifth.
- Open-ended straight draw: four cards in a row, like 4-5-6-7, which a card at either end completes (a 3 or an 8).
- Gutshot: four cards of a straight with a gap in the middle, like 4-5-7-8, which only a 6 completes.
- Overcards: hole cards higher than every card on the board, which could win by pairing.
- A bluff is a bet made with a weak hand to get better hands to fold.
The maths
Outs
'Outs' refer to all the possible cards that give you a strong chance of winning a given hand and - as we see in the next section - can be used to help estimate your chances of winning. To calculate your outs you simply count the number of cards that could come on the board and give you a winning 5 card hand. So for example if you get dealt
4s 5s
and the flop comes out as
4d 6h 7h
you already have a pair of 4s, and there are a few options to build stronger hands depending on what comes on the turn and river. So the next thing to do is think 'what single card could come next that would give me a strong hand?'. In terms of suits there's no opportunity for a flush (even if the turn and river both came as spades we would only have 4 of the 5 spades needed), which means we have to think about the rank of the cards.
The obvious one in this setup is going to be hitting a straight, as this gives us a really strong 5 card hand. If we re-arrange the cards based on rank then it'll make it a bit easier to see what we need next:
4d | 4s 5s 6h 7h
From this there are a few obvious cards that would help us
3c 3s 3h 3d
which makes a straight going from 3 up to 7, or we can have
8c 8s 8h 8d
which would give us a straight from 4 up to 8. None of these 8 cards have been seen yet, so that's 8 outs for the straight.
We can also improve our pair. Two of the four 4s are already showing (one in our hand and one on the flop), which leaves
4c 4h
Either of these would give us three 4s (trips), which is 2 more outs.
Adding these together gives us 4 + 4 + 2 = 10 outs. We can simply add them because no card is counted twice: a 4 doesn't complete our straight, and a 3 or an 8 doesn't give us three of a kind. So out of the 47 cards we haven't seen, 10 would give us a strong hand on the turn.
It's worth noting that not every out is equally good. The 3♥ and 8♥ would complete our straight but also put a third heart on the board, which could give an opponent holding two hearts a flush.
Pot equity
Your equity in a hand is your share of the pot, based on how often you'd win if all the remaining cards were dealt out. If you'd win 35% of the time you have 35% equity, so in a pot of 20bb your share is worth 7bb on average.
Outs are how we turn a hand into a probability. Because of the hearts, not all of our outs in the 4♠ 5♠ example are equally good, so to keep the maths clean we'll switch to a hand where every out is. Say you hold A♠ K♠ and the flop comes:
As Ks | Qs 7s 2d
You have a flush draw. There are 13 spades in the deck and 4 are already showing, so 9 spades are left that would complete your flush. You can see 5 cards (your 2 and the 3 on the board), so there are 47 cards you haven't seen. Pairing your ace or king could also win, but to keep things simple we'll only count the flush.
The chance of hitting your flush on the turn is:
$$\frac{9}{47} \approx 19.1%$$
The chance of hitting by the river, if you see both cards, is easiest to find by working out the chance of missing twice and taking it away from 1:
$$1 - \frac{38}{47} \times \frac{37}{46} \approx 35.0%$$
The rule of 2 and 4
Doing that maths at the table isn't exactly practical, so we can use a simple shortcut. If only the flop has been dealt, like in the example above, count your outs and multiply them by 4 to get an estimate of your pot equity. Once the turn has been dealt (so 4 community cards are showing), multiply your outs by 2 instead, since there's only one card to come. Our 9 flush outs give 36%, very close to the real 35.0%, and the 10 outs from the 4♠ 5♠ hand give 40% against a real 38.4%.
Multiplying by 4 assumes you'll get to see both the turn and the river. If there's more betting to come, multiplying by 2 is the safer estimate even on the flop, which we'll come back to in the expected value section.
| Draw |
Outs |
Next card |
By the river |
| Gutshot straight draw |
4 |
8.5% |
16.5% |
| Two overcards |
6 |
12.8% |
24.1% |
| Open-ended straight draw |
8 |
17.0% |
31.5% |
| Flush draw |
9 |
19.1% |
35.0% |
| Flush draw + open-ended straight draw |
15 |
31.9% |
54.1% |
The shortcut gets less accurate the more outs you have. For 15 outs it says 60% when the real figure is 54.1%, because it effectively counts the times you'd hit on both the turn and the river twice.
Pot odds
Equity tells you how often you'll win, whilst pot odds tell you the relative cost of staying in the hand, comparing how much you have to call with how much is already in the pot. They only cover the bet you're facing right now, though, not any betting still to come.
Say there's 10bb in the pot and your opponent bets 5bb. The pot is now 15bb and it costs you 5bb to call, so you're getting 15 to 5, or 3 to 1.
To compare that with your equity, turn it into the share of the final pot you're paying for:
$$\text{equity needed} = \frac{\text{call}}{\text{pot} + \text{call}} = \frac{5}{15 + 5} = 25%$$
Here the pot includes your opponent's bet. If you win more than 25% of the time, calling makes money in the long run.
The equity you need only depends on how big the bet is compared to the pot, so it's worth knowing the common sizes:
| Opponent's bet |
Equity needed to call |
| ¼ pot |
16.7% |
| ⅓ pot |
20% |
| ½ pot |
25% |
| ⅔ pot |
28.6% |
| Pot |
33.3% |
| 2× pot |
40% |
The bigger the bet, the more often you need to win, which is exactly why big bets put pressure on draws.
Expected value
The decision itself is a comparison: call when your equity is higher than the equity you need. The reason this works is expected value (EV), the amount you'd win or lose on average if you made the same call over and over:
$$\text{EV} = P(\text{win}) \times \text{pot} - P(\text{lose}) \times \text{call}$$
Back to the flush draw facing a bet of 5bb into 10bb. The pot is 15bb and the call is 5bb, so you need 25%:
- If your opponent is all-in, you'll see both the turn and the river, so your equity is 35.0%. $\text{EV} = 0.35 \times 15 - 0.65 \times 5 = +2\text{bb}$, so calling makes about 2bb on average. Call.
- If there's more betting to come, you can only count on seeing the next card, so your equity is 19.1%. If you give up whenever you miss, $\text{EV} \approx 0.191 \times 15 - 0.809 \times 5 \approx -1.2\text{bb}$. On pot odds alone, fold.
So the same hand can be a call or a fold depending on whether you get to see one card or two. Even then, the fold isn't always right: if you hit your flush on the turn, you'll often win more from your opponent on later streets, which pot odds don't count. That extra money is called implied odds.
All of this also assumes your outs are good, meaning you'd win whenever you hit. That depends on what your opponent actually has. Against a set of 7s, for example, the 2♠ completes your flush but also gives them a full house, so you really have 8 outs rather than 9. Your equity is really against the whole range of hands they could hold, which is where ranges come in.
Ranges
Ranges in poker have 2 purposes: the first is helping you establish what the other players at the table may have, and the second is acting as a rough guideline for how you should play your hand.
The aim of ranges is to provide a better way to make educated decisions on how to play in certain situations, however this only holds as an average rather than for hyper-specific scenarios. Say someone holding 2♠ 2♦ sees 2♣ 2♥ 5♥ come on the flop, giving them quad 2s. A range doesn't plan around hitting quads, because flops like that are incredibly rare and relying on them to win isn't sustainable in the long run.
Ranges can be represented in a variety of different ways, however the most common is as a matrix showing, given your current position (BTN, UTG, SB, etc.) and the current street (pre-flop, etc.), roughly what hands you should be playing based on statistics.
Predicting someone else's hand
The first use for a range in poker is to try and identify what hand a given player has based on how they actually play the game. Rather than putting them on one exact hand, you give them a range of possible hands and narrow it down as the hand goes on.
This starts before the flop. A player who raises from UTG still has the whole table left to act behind them, so they'll usually only do it with strong hands, such as pocket pairs and strong aces like AK and AQ. The same raise from the button could be a much wider range of hands.
Every action after that removes hands from their range. Say UTG raises, you call, and the flop comes:
Ks 7d 2c
If they bet big and keep betting on the turn, their range is weighted towards strong hands like AA, KK and AK. If they check and then fold to a bet, they probably had a pair like QQ or JJ that didn't like seeing the king.
This is also where the probability comes in, because the cards on the board change how many ways there are to hold each hand. A pocket pair normally has $\binom{4}{2} = 6$ combinations, but with a king on the board there are only 3 ways left to hold KK:
| Hand |
Combinations |
| AA |
6 |
| KK, 77 or 22 (a set) |
3 each, 9 in total |
| AK |
$4 \times 3 = 12$ |
So even though a set is the scary hand here, a player with this range is more likely to be holding AK than any set.
How should you play your hand
The second use is as a guide for your own play. Before the flop, this is what range charts are for: they show which hands to play from each position. The general rule is that the earlier you act, the tighter your range should be:
- Early position (UTG): there are more players behind you who could have a strong hand, and you'll often be out of position after the flop, so you only play roughly the top 15–20% of hands.
- Late position (CO, BTN): there are fewer players left to get past and you'll usually act last after the flop, so you can play far more hands, around 40% or more from the button.
After the flop, it helps to think about your own range the way your opponent would. If you only ever bet when you have a strong hand, a good player will quickly start folding whenever you bet. Playing your range means making the same play with a mix of hands, some strong and some bluffs, so your actions don't give your hand away.
Ranges also tell you when to be aggressive. The K♠ 7♦ 2♣ flop above favours the UTG raiser, because their range contains more AA, KK and AK than a caller's does. This is called having the range advantage, and it means UTG can bet confidently even with a weaker hand, because their range as a whole is ahead.
Finally, ranges are a starting point rather than a rulebook. Against a player who raises far more hands than they should, widen the range you give them; against someone who only plays the strongest hands, tighten it.
Wrapping up
Going back to the start, the reason this is worth learning is how closely it maps onto trading:
- Expected value is your edge. A call with positive EV is like a trade with a positive expected return: it can still lose, and judging it on one result tells you very little. A +2bb call that misses was still the right call.
- Variance is why the long run matters. Just like the S&P 500's 10% average hides its bad years, a winning strategy still has losing sessions. The law of large numbers only helps if you stay in the game long enough to reach the long run.
- Stacks in big blinds are position sizing. Measuring your stack in big blinds is the same idea as sizing a position relative to your capital rather than in absolute terms: what matters is how much of what you have is at risk.
- Ranges are thinking in distributions. You never know your opponent's exact hand, just like you never know exactly who's on the other side of a trade, so you reason about the whole range of possibilities and update it as new information comes in.
None of this guarantees you'll win any single hand. But making decisions with positive expected value, and sizing them so that a bad run can't knock you out, is how you come out ahead over time, at the table or in the markets.
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