Chance to hit by the river
44.96%
12 outs, two cards to come
- Chance on the next card25.53%
- Odds against by the river1.22 : 1
- Rule of 4 estimate+3.04 points from the exact chance48%
The count behind 12 outs on the flop
A 44.96% chance is the territory of big draws. The miss is 55.04%, so even a bet the size of the pot can be a call.
- Unseen cards47
- Your outs12
- Cards that miss35
- Pairs of cards with no out595
- Pairs with at least one out486
- Chance by the river44.96%
- Chance on the next card25.53%
- Odds against hitting1.22 : 1
- Estimate from the rule of 448%
- Error of the rule of 4+3.04 points
- Adjusted rule 4k - (k - 8)44%
- Error of the adjusted rule-0.96 points
Chance by the river = 1 - C(u - k, 2) / C(u, 2), with u = 47 unseen cardsFor 12 outs: 1 - C(35, 2) / C(47, 2) = 1 - 595 / 1,081 = 44.96%Next card = 12 / 47 = 25.53%
Start with the cards you have not seen: 47 of them. 35 of those miss and 12 are outs. Two cards come, so there are 1,081 possible pairs, and 595 of them hold no out at all.
A 48% estimate against a true 44.96% is an overstatement of 3.04 points, more than the 2 points that make a shortcut safe to use unchecked.
For counts over 8, the rule 4k - (k - 8) is the one to remember. Here it gives 44% with an error of -0.96 points.
Which holding has 12 outs?
One standard source of this count is a flush draw with a gutshot. Count the cards carefully at the table, because outs that also help a rival do not count in full.
A shared out has a price: 3.24 points for the first, and 9.99 points in all with 3 shared. Subtract any shared outs before you compare with a bet.
| Shared outs | Effective outs | Chance by the river | Chance given up |
|---|---|---|---|
| 0 | 12 | 44.96% | 0.00 points |
| 1 | 11 | 41.72% | 3.24 points |
| 2 | 10 | 38.39% | 6.57 points |
| 3 | 9 | 34.97% | 9.99 points |
The bets 12 outs can call on the flop
Compare the columns: required equity on the left, margin in the middle, verdict at the end. With 12 outs you can call all 12 bet sizes.
Work the price backward from 44.96% and you find a limit of 445.87% of the pot: $445.87 into a $100 pot.
| Bet size | Equity a call needs | Margin over it | Verdict |
|---|---|---|---|
| 1/4 pot | 16.67% | +28.29 points | Call |
| 1/3 pot | 20.00% | +24.96 points | Call |
| 2/5 pot | 22.22% | +22.74 points | Call |
| 1/2 pot | 25.00% | +19.96 points | Call |
| 2/3 pot | 28.57% | +16.39 points | Call |
| 3/4 pot | 30.00% | +14.96 points | Call |
| full pot | 33.33% | +11.63 points | Call |
| 1.25x pot | 35.71% | +9.24 points | Call |
| 1.5x pot | 37.50% | +7.46 points | Call |
| 2x pot | 40.00% | +4.96 points | Call |
| 3x pot | 42.86% | +2.10 points | Call |
| 4x pot | 44.44% | +0.51 points | Call |
Nearby counts on both streets
Counts near 12 sit side by side below, on the flop and on the other street. One out is worth around 3.15 points on this page.
- 12 outs on the turn: 26.09% on the river.
- 11 outs on the flop: 41.72% by the river.
- 13 outs on the flop: 48.10% by the river.
When a bet buys only one card
Two views, one draw: 44.96% if you see both cards and 25.53% if you see one. Against 25.00%, the one-card view has a margin of +0.53 points and reads Call.
Only 4 bet sizes stay calls on the next card alone, against 12 on the two-card figure.
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 12 outs that is 1 - 595 / 1,081 = 44.96%. The chance on the next card is k / 47, here 25.53%.
The rule of 4 figure is 4 x k, here 48%, and its error is that figure minus the exact chance in percentage points, here +3.04 points. The adjusted rule 4k - (k - 8) gives 44%, an error of -0.96 points.
The model gives every one of the 47 unseen cards the same chance and removes nothing for what a rival holds. It measures hitting an out, not winning the hand. Bets are priced on a $100 pot, and a call adds one bet to the pot.
Computed from the closed form by the GeorgeLiPoker math library.
Updated
Questions
Out of 100 hands, how many hit with 12 outs on the flop?
With 12 outs on the flop you hit by the river 44.96% of the time, about 45 of every 100 hands, and on the next card 25.53%. The odds against hitting by the river are 1.22 : 1, since 595 of 1,081 pairs of unseen cards hold no out.
How does the quick rule compare with the exact chance for 12 outs?
The rule of 4 gives 48% for 12 outs against an exact 44.96%, an error of +3.04 points, so it overstates the draw. The adjusted rule 4k - (k - 8) gives 44%, an error of -0.96 points, and is the closer figure.
What is the largest bet worth calling with a count of 12 on the flop?
At 44.96% by the river, you can call all 12 bet sizes. A half-pot bet needs 25.00%, a margin of +19.96 points, so against half the pot the verdict is Call. The largest bet you can call is 445.87% of the pot, or $445.87 on a $100 pot.
How many points do I lose if outs are shared with a better hand?
Take them off the count first. With one shared out the chance by the river falls from 44.96% to 41.72%, and with 3 shared it falls to 34.97%, a loss of 9.99 points. The effective count is then 9 outs.
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