Count your outs
Enter the cards that improve your hand and pick the street.
Chance to hit by the river
34.97%
- Next card only19.15%
- Rule of 4Its error: +1.03 points36%
- Adjusted rule of 4Its error: +0.03 points35%
- Effective outs9
- Unseen cards47
What these numbers assume:
- Every unseen card is equally likely.
- No cards are removed for your opponent's hand.
- The chance by the river assumes you see both cards.
How the outs math works
The formulas
An out is an unseen card that would improve your hand to one you expect to win. You count your outs, then divide by the cards you have not seen. On the flop you have seen five cards (your two and three on the board), so 47 remain. On the turn you have seen six, so 46 remain.
Unseen cards u = 47 on the flop, 46 on the turnEffective outs k' = k - d, where k is your outs and d is the number shared with a better handNext card: k' / uFlop to river: p = 1 - C(u - k', 2) / C(u, 2)Turn to river: p = k' / 46Equity a call needs: e = C / (P + B + C), with pot P, bet B and call C
The rule of 4 estimates the chance to hit by the river from the flop as 4 times your outs, and the rule of 2 estimates the chance on one card as 2 times your outs. With 9 outs the rule of 4 gives 36% against an exact 34.97%, which is +1.03 points. With 15 outs it gives 60% against 54.12%, which is +5.88 points.
Above 8 outs the rule runs high, so a common fix is to take off the number of outs above 8: 4k' - (k' - 8). With 15 outs that gives 53%, which is -1.12 points from the exact figure.
A worked example
You hold a flush draw on the flop, 9 outs, and the pot is $100. Your opponent bets $50, half the pot, so a call costs $50.
Chance to hit by the river: 34.97%. Chance on the next card: 19.15%. Equity a call needs: $50 / ($100 + $50 + $50) = 25.00%.
Against the two-card figure the call is +9.97 points above break-even, so the verdict is: Call is profitable. Against the next card alone it is -5.85 points, which is why it matters whether more betting is still to come.
Using the calculator
- Enter the number of outs, the cards that would improve your hand. Tap a shortcut such as a flush draw to fill in the usual count.
- Choose the flop if two cards are still to come, or the turn if one card is to come.
- If some of your outs would also complete a better hand for an opponent, enter how many. They are taken off before the chances are worked out.
- To compare with a bet, enter the pot before the bet and the size of the bet. The calculator shows the equity a call needs and how far your chance is above or below it.
Common mistakes
- Counting outs that also improve your opponent. If a card gives you a flush but gives them a full house, it should not count in full.
- Using the two-card figure when you face a bet and more betting is still to come on the turn. In that spot you pay for the next card, so compare the next-card chance with the price.
- Applying the rule of 4 with a large number of outs. It overstates the chance more and more as the count goes up.
How this is computed
The method
Every figure is the exact closed form, with no simulation and no random sampling. The chance to hit by the river from the flop is one minus the chance that both cards miss: C(u - k', 2) pairs of cards contain no out, out of C(u, 2) possible pairs, where u is 47 and k' is your outs after the shared ones are taken off.
The chance on one card is k' divided by the unseen cards. The rule figures are 4k' on the flop and 2k' on the turn, shown beside the exact figure with the difference in percentage points.
The comparison with a bet uses the equity a call needs, call / (pot + bet + call), and checks your chance to hit by the river against it. The calculator assumes every unseen card is equally likely and removes no cards for an opponent, so it gives the chance to hit your outs, not your equity against a particular hand.
Chance to hit, outs 1 to 20
| Outs | Flop, by the river | Flop, next card | Turn, on the river | Rule of 4 error |
|---|---|---|---|---|
| 1 | 4.26% | 2.13% | 2.17% | -0.26 points |
| 2 | 8.42% | 4.26% | 4.35% | -0.42 points |
| 3 | 12.49% | 6.38% | 6.52% | -0.49 points |
| 4 | 16.47% | 8.51% | 8.70% | -0.47 points |
| 5 | 20.35% | 10.64% | 10.87% | -0.35 points |
| 6 | 24.14% | 12.77% | 13.04% | -0.14 points |
| 7 | 27.84% | 14.89% | 15.22% | +0.16 points |
| 8 | 31.45% | 17.02% | 17.39% | +0.55 points |
| 9 | 34.97% | 19.15% | 19.57% | +1.03 points |
| 10 | 38.39% | 21.28% | 21.74% | +1.61 points |
| 11 | 41.72% | 23.40% | 23.91% | +2.28 points |
| 12 | 44.96% | 25.53% | 26.09% | +3.04 points |
| 13 | 48.10% | 27.66% | 28.26% | +3.90 points |
| 14 | 51.16% | 29.79% | 30.43% | +4.84 points |
| 15 | 54.12% | 31.91% | 32.61% | +5.88 points |
| 16 | 56.98% | 34.04% | 34.78% | +7.02 points |
| 17 | 59.76% | 36.17% | 36.96% | +8.24 points |
| 18 | 62.44% | 38.30% | 39.13% | +9.56 points |
| 19 | 65.03% | 40.43% | 41.30% | +10.97 points |
| 20 | 67.53% | 42.55% | 43.48% | +12.47 points |
Computed from the closed form by the GeorgeLiPoker math library.
Questions
How many outs does a flush draw have?
A flush draw has 9 outs. You can see four cards of the suit between your hand and the board, and 13 minus 4 leaves 9 of that suit unseen. On the flop that is 34.97% to complete by the river and 19.15% on the next card. On the turn it is 19.57% on the river.
What is the rule of 2 and 4, and how far off is it?
Multiply your outs by 4 on the flop to estimate the chance to hit by the river, or by 2 with one card to come. With 9 outs the rule of 4 says 36% against an exact 34.97%, +1.03 points. The error grows with more outs: 15 outs reads 60% against 54.12%.
Should I count all my outs?
Not always. An out that also completes a better hand for an opponent can still lose you the pot, so it should not count in full. Enter those cards as outs that also help a better hand and the calculator takes them off first. With 9 outs and 2 shared, the chance to hit by the river from the flop falls from 34.97% to 27.84%.
Why use the next-card chance when I face a bet on the flop?
A bet on the flop only buys you the next card. If another bet is likely on the turn, you pay again for the river, so the fair comparison is the next-card chance, 19.15% with 9 outs, against the equity the call needs. The by-the-river figure fits when you are all-in or when no more betting will follow.
Does the calculator account for my opponent's cards?
No. It assumes all 47 unseen cards on the flop are equally likely and removes no cards for an opponent's hand or range, because from your seat their hole cards are unseen like the rest of the deck. The result is the chance to hit your outs, not your equity against a named hand. For that, use the equity calculator.
Related pages
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