# Pot Odds and Outs: The Only Poker Maths You Need

How to count outs, what the rule of 4 and 2 gets wrong, and how to turn a pot size and a bet into the exact percentage you need to win. With the full outs chart.

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- **Canonical URL:** https://dothecalculation.com/blog/hobby/pot-odds-and-outs
- **Category:** Hobbies & Leisure Utilities
- **Author:** Do The Calculation Team
- **Published:** 2026-09-20
- **Reading time:** 16 min read
- **Publisher:** Do The Calculation (https://dothecalculation.com)
- **Methodology:** https://dothecalculation.com/methodology

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This is a reference for the probability and combinatorics behind drawing hands — the same maths taught in any poker strategy book. It is not advice to gamble, and nothing here suggests risking money you would rather keep.

With that said, there is a single decision that comes up more than any other at a poker table: someone has bet, you have a draw, and you need to know whether calling is mathematically sound. It takes two numbers and one comparison, and you can do it in your head in about five seconds.

> **The comparison** — Work out how often you will complete your draw. Work out what share of the final pot you are being asked to put in. If the first number is bigger than the second, calling is profitable on this street's maths. That is the whole framework.

## Step one: count your outs

An out is a card still unseen that completes your hand. Counting them is the part people get wrong, usually by counting cards that would give them a hand that still loses.

- Flush draw: four to a flush means nine of the thirteen suited cards are left. Nine outs.
- Open-ended straight draw: two different ranks complete it, four of each. Eight outs.
- Gutshot: one rank completes it. Four outs.
- Pocket pair hoping to hit a set: two cards of that rank remain. Two outs.
- Two overcards, hoping either pairs: three of each. Six outs — and this is the one to be careful with, because pairing an overcard often does not win.

> **Discount outs that do not win** — If you hold four to a flush and the board is paired, some of your flush cards arrive on a board that has already made someone a full house. Those are not outs. Counting nine when three of them lose is how a correct-looking calculation produces a losing call. Experienced players talk about "clean" outs for exactly this reason.

## Step two: turn outs into a percentage

The exact figure is a hypergeometric calculation: one minus the probability that none of your outs arrives. With two cards to come from 47 unseen, that is 1 − C(47 − outs, 2) ÷ C(47, 2).

**The outs chart — exact probabilities against the shortcut**
| Outs | Typical draw | Turn + river (exact) | ×4 rule | River only (exact) | ×2 rule |
| --- | --- | --- | --- | --- | --- |
| 2 | Pocket pair to a set | 8.4% | 8% | 4.3% | 4% |
| 4 | Gutshot straight | 16.5% | 16% | 8.7% | 8% |
| 6 | Two overcards | 24.1% | 24% | 13.0% | 12% |
| 8 | Open-ended straight | 31.5% | 32% | 17.4% | 16% |
| 9 | Flush draw | 35.0% | 36% | 19.6% | 18% |
| 12 | Flush + gutshot | 45.0% | 48% | 26.1% | 24% |
| 15 | Flush + open-ended | 54.1% | 60% | 32.6% | 30% |

The rule of 4 and 2 — multiply outs by 4 with two cards to come, by 2 with one — is the mental shortcut, and the table shows exactly where it is safe. Up to about nine outs it is within a point or so, which is fine at a table. Past that it drifts badly: at fifteen outs it claims 60% when the truth is 54.1%, an overstatement of nearly six percentage points.

That matters because big combo draws are precisely the hands where large amounts of money go in. If you are using the shortcut to justify a stack-sized decision on a fifteen-out draw, you are using the one number it gets worst.

Tool: [Exact odds for any number of outs](https://dothecalculation.com/calculators/poker-odds-calculator) — Hypergeometric probability for your outs and cards to come, with the rule-of-thumb figure shown alongside so you can see the gap.

## Step three: what the pot is offering

Pot odds answer one question: what percentage of the time do you need to win for calling to break even? It depends only on the money already in the pot and the bet in front of you.

**Required equity**

```
Required equity = bet to call ÷ (pot + bet to call)
```
- The denominator includes your own call, because that money is part of the pot you are trying to win.
- Expressed as odds instead, it is pot ÷ bet to one. A $100 pot and a $20 bet is 5:1, which is the same statement as 16.7%.

**Common bet sizes**
| Pot | Bet to call | Odds | You need to win |
| --- | --- | --- | --- |
| $200 | $25 | 8:1 | 11.1% |
| $100 | $20 | 5:1 | 16.7% |
| $100 | $50 | 2:1 | 33.3% |
| $100 | $100 | 1:1 | 50.0% |

Notice how fast the requirement climbs. A quarter-pot bet needs you to win one time in six; a pot-sized bet needs you to win half the time. This is why bet sizing is itself a strategic decision rather than a habit.

## Putting the two together

**Two worked decisions**
| Situation | Your equity | Required | Verdict |
| --- | --- | --- | --- |
| Flush draw, 9 outs, 2 cards to come, $20 into $100 | 35.0% | 16.7% | Call — 18.3 points clear |
| Gutshot, 4 outs, 1 card to come, $25 into $50 | 8.7% | 33.3% | Fold — 24.6 points short |

The first is a comfortable call and the second is a clear fold, and the gap in both cases is large enough that you do not need to be precise about the decimals. Most real decisions are like this. The genuinely marginal ones are rarer than people think, and when a decision is marginal on pot odds the other factors below decide it.

## What this calculation leaves out

- Implied odds — money you expect to win on later streets if you hit. A draw that is slightly short on pot odds can still be a call if hitting it will get paid.
- Reverse implied odds — money you lose when you hit and still lose the hand. Drawing to the low end of a straight is the classic case.
- Whether there is more betting to come. The two-cards-to-come figure assumes you get to see both cards, which is not true if your opponent will bet again on the turn.
- Fold equity. Raising can win the pot immediately, which is a different calculation entirely.
- Whether your outs are clean. See above — this one is on you, not the formula.

> **The two-cards-to-come trap** — Multiplying by 4 gives the chance of hitting across both the turn and the river, and it is only the right number if you are seeing both cards for the price in front of you — typically when you or your opponent is already all in. If there is another betting round coming, you are usually paying to see one card, so the ×2 column is the honest one.

## Counting outs when draws overlap

The outs chart lists a combo draw of a flush draw plus an open-ended straight draw as 15 outs. That number is worth deriving once, because the method is how you count any combination correctly, and the naive answer is wrong.

- A flush draw has 9 outs: the 9 remaining cards of your suit.
- An open-ended straight draw has 8 outs: 4 cards of each of the two ranks that complete it.
- Adding them gives 17, which double counts. Two of the straight cards are also of your suit, one of each completing rank, and they are already among the 9 flush outs.
- So the true count is 9 + 8 − 2 = 15.

The same subtraction applies to every combination: add the outs for each draw, then subtract any card that completes more than one of them. Get into the habit and you will stop overstating combo draws, which, as the chart shows, are already the hands where the shortcut rule overstates your chances the most.

## Before the flop: what you are dealt

Outs start to matter on the flop, but the same counting method explains the probabilities that come before it. There are C(52, 2) = 1,326 possible two-card starting hands.

**Starting hand probabilities, exact**
| Starting hand | Combinations | Probability |
| --- | --- | --- |
| Any pocket pair | 78 | 5.88% |
| A specific pair, such as two aces | 6 | 0.45% |
| Any two suited cards | 312 | 23.53% |
| Ace-king, suited or not | 16 | 1.21% |
| Ace-king suited | 4 | 0.30% |

A specific pair such as aces arrives about once every 221 hands. Any pocket pair arrives about once every 17. Those frequencies are useful less for strategy than for calibrating your sense of what is rare: players who feel they are dealt premium hands too rarely are usually comparing their memory with an impression, not with 0.45%.

## What the flop does for you

The flop is three cards from the 50 you cannot see, so there are C(50, 3) = 19,600 possible flops. Here is how often it gives you the draws and hands this guide is about.

**Flop probabilities, exact**
| You hold | The flop gives you | Probability |
| --- | --- | --- |
| A pocket pair | A set or better (at least one card of your rank) | 11.76% |
| Two suited cards | A flush draw (exactly two more of your suit) | 10.94% |
| Two suited cards | A made flush (three of your suit) | 0.84% |
| Two unpaired cards | At least one card pairing one of them | 32.43% |

The pocket pair figure is the one worth remembering. You flop a set or better a little under one time in eight, which is why calling a large pre-flop raise with a small pair hoping to hit a set only makes sense when the amount you can win when you do hit is large compared with the call. That is implied odds in its purest form, and it is covered below.

## Expected value: how much a call is worth

Pot odds tell you whether a call breaks even. Expected value tells you how much it makes or loses on average, which is more useful when you are comparing two choices that are both profitable, or both not.

**Expected value of a call, on this street's money only**

```
EV = (equity × pot) − ((1 − equity) × bet to call)
```
- Pot here means everything already in the middle, including your opponent's bet, but not your own call.
- A positive EV means the call wins money on average over many repetitions; a negative one means it loses.

**The worked decisions from above, as expected value**
| Situation | Equity | Expected value of calling |
| --- | --- | --- |
| Flush draw, $20 into a $100 pot, both cards to come (all in) | 34.97% | +$21.96 |
| Flush draw, $20 into a $100 pot, river only | 19.57% | +$3.48 |
| Gutshot, $25 into a $50 pot, river only | 8.70% | −$18.48 |

The middle row is the instructive one. On pot odds alone the river-only flush draw is still a call, since 19.57% beats the 16.7% required, but the margin is thin: about $3.48 per call on average. The two-card version is worth more than six times as much, which is exactly the difference between the ×4 and ×2 columns of the chart, now measured in money.

## Implied odds, put into numbers

Implied odds is usually described vaguely, as money you might win later. It can be made precise by asking a single question: how much more would you have to win, on average, when you hit, for this call to break even?

**Extra winnings needed to justify a call**

```
Extra needed when you hit = (bet to call ÷ equity) − (pot + bet to call)
```
- Equity here is your chance of hitting on the next card only.
- If the answer is small compared with what your opponent is likely to pay you after you hit, the call can be justified; if it is large, it cannot.

**How much more you would need to win when the draw comes in**
| Situation | Pot odds verdict | Extra needed when you hit |
| --- | --- | --- |
| Gutshot, $25 into $50, one card to come | Fold: 8.7% against 33.3% needed | About $213 more |
| Flush draw, $50 into $100, one card to come | Fold: 19.6% against 33.3% needed | About $106 more |

For the gutshot, you would need to win about $213 more on the later street, on average, every time you hit, to make up for a pot that is only $75. That is almost three times the current pot, which is rarely realistic. The flush draw facing a larger bet needs about $106 more, roughly the size of the existing pot, which is sometimes achievable against an opponent who will keep betting. This is the honest version of implied odds: a number you can compare with what you actually expect to be paid, rather than a feeling that you might get paid.

> **Reverse implied odds cut the other way** — The calculation above assumes you win whenever you hit. If hitting your draw sometimes gives you the second-best hand, such as the low end of a straight when a higher one is possible, you lose extra money in exactly the situations you were counting on. Subtract those cases before relying on implied odds.

## Why correct calls still lose often

A mathematically correct call is not a call that usually wins. It is a call that makes money over many repetitions. Even the good draws in this guide miss most of the time, and they miss in streaks more often than intuition expects.

**How often a draw misses several times in a row**
| Draw | Chance of missing 3 times in a row | Chance of missing 5 times in a row |
| --- | --- | --- |
| Flush draw, river only (19.6%) | 52.0% | 33.6% |
| Flush draw, turn and river (35.0%) | 27.5% | 11.6% |

A river-only flush draw misses three in a row more often than not. Seeing that happen is not evidence that the maths is wrong or that the deck is against you; it is the expected behaviour of a 19.6% event. This is one of the most useful things the numbers can do for a recreational player: they separate a bad decision from a bad outcome, and they are a good reason to decide in advance how much you are prepared to lose in a session.

## Backdoor draws: counting runner-runner outs

A backdoor draw needs both the turn and the river to help. The classic example is holding three cards to a flush after the flop, needing two more of your suit in a row. With 10 cards of your suit left among 47 unseen, the chance is C(10, 2) ÷ C(47, 2) = 45 ÷ 1,081 = 4.16%.

That is why experienced players count a backdoor flush draw as roughly one extra out: it adds about four percentage points to a hand's equity over two cards, the same as one clean out would under the ×4 rule. It is rarely enough to call with on its own, but it can tip a close decision, and it is a good example of how small probabilities combine when you count them properly.

## Common mistakes with poker probability

- **Double counting combo draws.** Subtract cards that complete more than one draw. A flush draw plus an open-ended straight draw is 15 outs, not 17.
- **Using the two-card figure when only one card is guaranteed.** Unless someone is all in, you are usually paying for one card at a time.
- **Counting outs that do not win.** A card that completes your draw but gives an opponent a better hand is not an out.
- **Treating implied odds as a feeling.** Work out the extra you would need to win and ask honestly whether you will be paid that much.
- **Judging a decision by one result.** A 35% draw that misses was still a correct call if the price was right, and a 9% draw that hits was still a bad one.

## The complete outs chart, 1 to 20 outs

The chart earlier in this guide covers the common draws. Here is every count from 1 to 20 outs, so you can look up unusual combinations directly. The fourth column shows how far the ×4 shortcut is from the truth with two cards to come; positive means the shortcut understates your chances, negative means it overstates them.

**Exact probability of hitting at least one out, 47 unseen cards after the flop**
| Outs | Turn and river (exact) | ×4 rule | Shortcut error | River only (exact) | ×2 rule |
| --- | --- | --- | --- | --- | --- |
| 1 | 4.3% | 4% | +0.3 | 2.2% | 2% |
| 2 | 8.4% | 8% | +0.4 | 4.3% | 4% |
| 3 | 12.5% | 12% | +0.5 | 6.5% | 6% |
| 4 | 16.5% | 16% | +0.5 | 8.7% | 8% |
| 5 | 20.4% | 20% | +0.4 | 10.9% | 10% |
| 6 | 24.1% | 24% | +0.1 | 13.0% | 12% |
| 7 | 27.8% | 28% | -0.2 | 15.2% | 14% |
| 8 | 31.5% | 32% | -0.5 | 17.4% | 16% |
| 9 | 35.0% | 36% | -1.0 | 19.6% | 18% |
| 10 | 38.4% | 40% | -1.6 | 21.7% | 20% |
| 11 | 41.7% | 44% | -2.3 | 23.9% | 22% |
| 12 | 45.0% | 48% | -3.0 | 26.1% | 24% |
| 13 | 48.1% | 52% | -3.9 | 28.3% | 26% |
| 14 | 51.2% | 56% | -4.8 | 30.4% | 28% |
| 15 | 54.1% | 60% | -5.9 | 32.6% | 30% |
| 16 | 57.0% | 64% | -7.0 | 34.8% | 32% |
| 17 | 59.8% | 68% | -8.2 | 37.0% | 34% |
| 18 | 62.4% | 72% | -9.6 | 39.1% | 36% |
| 19 | 65.0% | 76% | -11.0 | 41.3% | 38% |
| 20 | 67.5% | 80% | -12.5 | 43.5% | 40% |

The error column tells the whole story of the shortcut. It is within about one point up to 9 outs, reaches about three points at 12 outs and about six at 15, and keeps growing after that, to 12.5 points at 20 outs. The ×2 rule for a single card behaves better, because one card is a simple fraction, outs divided by 46, and ×2 understates it only gradually, by about 3.5 points even at 20 outs. If you remember one thing from this table, make it this: the ×4 rule is a small-draw tool, and on big combo draws you should use the exact figure.

## Sources

- Every probability in this guide is a direct hypergeometric calculation on a 52-card deck, computed from the formula shown rather than quoted from a chart. With two cards to come the unseen count is 47 (52 minus two hole cards and three flop cards), and with one card to come it is 46.

## Where to go next

The same expected-value reasoning underpins rating systems. [How Elo ratings work](/blog/hobby/elo-rating-explained) shows how a predicted result and an actual result combine into a rating change, which is pot odds with the money replaced by rating points.

## Common questions

**How do I calculate pot odds?**

Divide the bet you must call by the pot plus that bet. Calling $20 into a $100 pot means $20 ÷ $120 = 16.7%, so you need to win at least 16.7% of the time to break even. Stated as odds, that is 5:1.

**What are the odds of hitting a flush draw?**

35.0% with both the turn and river to come, and 19.6% with only the river. Those are the exact figures for nine outs; the rule of 4 and 2 gives 36% and 18%, which is close enough to use at a table.

**Is the rule of 4 and 2 accurate?**

Up to about nine outs, yes — within roughly a percentage point. Beyond that it overstates your equity, and badly: at fifteen outs it says 60% when the true figure is 54.1%. Those large combo draws are exactly the hands where the money is biggest, so use the exact number there.

**How many outs does an open-ended straight draw have?**

Eight — two ranks complete the straight and there are four of each. That is 31.5% with two cards to come and 17.4% with one. A gutshot has four outs, at 16.5% and 8.7%.

**What are implied odds?**

The money you expect to win on later streets if your draw comes in, on top of what is in the pot now. They can justify a call that pot odds alone do not, but they rely on a prediction about your opponent rather than on arithmetic, so they are much easier to talk yourself into than to actually collect.

**Should I use the turn-and-river number or the river-only number?**

River-only, unless you are seeing both cards for the price currently in front of you — which usually means someone is all in. If another betting round is coming, you are paying for one card at a time, and using the two-card figure overstates what your call is buying.

**How many outs is a flush draw plus a straight draw?**

Fifteen for a flush draw plus an open-ended straight draw: 9 flush outs plus 8 straight outs, minus the 2 straight cards that are also of your suit and already counted. That is 54.1% with two cards to come and 32.6% with one.

**What are the odds of flopping a set?**

With a pocket pair, 11.76% to flop at least one more card of your rank, a little under one time in eight. That is why calling a large raise with a small pair only makes sense when you can expect to win a lot when you do hit.

**How do I calculate the expected value of a call?**

Multiply your equity by the pot, which includes your opponent's bet, and subtract the chance of missing multiplied by your call. A flush draw calling $20 into a $100 pot with one card to come is 0.1957 × 100 − 0.8043 × 20, about +$3.48.

**How do I work out implied odds?**

Divide your call by your chance of hitting on the next card, then subtract the pot plus your call. The result is how much extra you would need to win, on average, each time you hit. A gutshot calling $25 into $50 needs about $213 more, which is rarely realistic.

**What is a backdoor flush draw worth?**

Needing two more cards of your suit on the turn and river, with 10 left among 47 unseen, is C(10,2) ÷ C(47,2) = 4.16%. Players often count it as roughly one extra out.

**Up to how many outs is the rule of 4 accurate?**

Within about one percentage point up to 9 outs. By 12 outs it overstates your chance by about 3 points, by 15 outs by about 6, and by 20 outs by 12.5. The rule of 2 for a single card stays closer, because one card is simply outs divided by 46; it understates slightly, by about 3.5 points at 20 outs.

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_Source: [Do The Calculation](https://dothecalculation.com/blog/hobby/pot-odds-and-outs). Quote freely with attribution and a link to this page._