Chance to hit by the river
54.12%
15 outs, two cards to come
- Chance on the next card31.91%
- Odds against by the river0.85 : 1
- Rule of 4 estimate+5.88 points from the exact chance60%
The count behind 15 outs on the flop
A 54.12% chance is the territory of big draws. The miss is 45.88%, so even a bet the size of the pot can be a call.
- Unseen cards47
- Your outs15
- Cards that miss32
- Pairs of cards with no out496
- Pairs with at least one out585
- Chance by the river54.12%
- Chance on the next card31.91%
- Odds against hitting0.85 : 1
- Estimate from the rule of 460%
- Error of the rule of 4+5.88 points
- Adjusted rule 4k - (k - 8)53%
- Error of the adjusted rule-1.12 points
Chance by the river = 1 - C(u - k, 2) / C(u, 2), with u = 47 unseen cardsFor 15 outs: 1 - C(32, 2) / C(47, 2) = 1 - 496 / 1,081 = 54.12%Next card = 15 / 47 = 31.91%
Think in pairs. The 47 unseen cards form 1,081 pairs, and only 496 avoid every out. Subtract and you land on 585 helpful pairs, a 54.12% chance.
The rule of 4 overshoots by 5.88 points with 15 outs. That is why an adjusted rule exists, and why the exact figure comes first on this page.
Use the adjusted rule when you count more than 8 outs on the flop. For 15 outs it reads 53%, 1.12 points away from 54.12%.
Which holding has 15 outs?
This count has a familiar home: a flush draw with an open-ended straight draw. It is worth knowing, because a quick look at the board often tells you the count.
Count carefully: a card that gives you the draw and gives an opponent a better hand should come off the list. One shared out takes the chance down by 2.96 points, to 51.16%.
| Shared outs | Effective outs | Chance by the river | Chance given up |
|---|---|---|---|
| 0 | 15 | 54.12% | 0.00 points |
| 1 | 14 | 51.16% | 2.96 points |
| 2 | 13 | 48.10% | 6.01 points |
| 3 | 12 | 44.96% | 9.16 points |
The bets 15 outs can call on the flop
Each row compares your 54.12% chance with the equity a call needs. You can call all 12 bet sizes with 15 outs on the flop.
With 54.12% you can call any size of bet. Even an enormous bet only needs a bit under 50% equity, and you hold more than that.
| Bet size | Equity a call needs | Margin over it | Verdict |
|---|---|---|---|
| 1/4 pot | 16.67% | +37.45 points | Call |
| 1/3 pot | 20.00% | +34.12 points | Call |
| 2/5 pot | 22.22% | +31.89 points | Call |
| 1/2 pot | 25.00% | +29.12 points | Call |
| 2/3 pot | 28.57% | +25.55 points | Call |
| 3/4 pot | 30.00% | +24.12 points | Call |
| full pot | 33.33% | +20.78 points | Call |
| 1.25x pot | 35.71% | +18.40 points | Call |
| 1.5x pot | 37.50% | +16.62 points | Call |
| 2x pot | 40.00% | +14.12 points | Call |
| 3x pot | 42.86% | +11.26 points | Call |
| 4x pot | 44.44% | +9.67 points | Call |
Nearby counts on both streets
Use the rows below to see what the count next door looks like. A step of one out is about 2.87 points on the flop.
- 15 outs on the turn: 32.61% on the river.
- 14 outs on the flop: 51.16% by the river.
- 16 outs on the flop: 56.98% by the river.
When a bet buys only one card
Two views, one draw: 54.12% if you see both cards and 31.91% if you see one. Against 25.00%, the one-card view has a margin of +6.91 points and reads Call.
Counting the next card alone, you can call 6 of the 12 bet sizes, compared with all 12 bet sizes when you see both cards.
How this is computed
The chance to hit by the river from the flop is 1 - C(u - k, 2) / C(u, 2), where u = 47 unseen cards and k is your effective outs. For 15 outs that is 1 - 496 / 1,081 = 54.12%. The chance on the next card is k / 47, here 31.91%.
The rule of 4 figure is 4 x k, here 60%, and its error is that figure minus the exact chance in percentage points, here +5.88 points. The adjusted rule 4k - (k - 8) gives 53%, an error of -1.12 points.
Read the figures as a card count. All 47 unseen cards stay in, including any your opponent holds, so the figures describe your outs, not a matchup. Bets are called in full against a $100 pot with no extra money assumed.
Computed from the closed form by the GeorgeLiPoker math library.
Updated
Questions
What percent chance does a count of 15 give you on the flop?
With 15 outs on the flop you hit by the river 54.12% of the time, about 54 of every 100 hands, and on the next card 31.91%. The odds against hitting by the river are 0.85 : 1, since 496 of 1,081 pairs of unseen cards hold no out.
Can I trust the shortcut for 15 outs on the flop?
The rule of 4 gives 60% for 15 outs against an exact 54.12%, an error of +5.88 points, so it overstates the draw. The adjusted rule 4k - (k - 8) gives 53%, an error of -1.12 points, and is the closer figure.
What is the largest bet worth calling with a count of 15 on the flop?
At 54.12% by the river, you can call all 12 bet sizes. A half-pot bet needs 25.00%, a margin of +29.12 points, so against half the pot the verdict is Call. Any bet size is a call, because 54.12% is above one half.
What is the effective count if some of the 15 outs are shared?
Take them off the count first. With one shared out the chance by the river falls from 54.12% to 51.16%, and with 3 shared it falls to 44.96%, a loss of 9.16 points. The effective count is then 12 outs.
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