Chance to hit by the river
12.49%
3 outs, two cards to come
- Chance on the next card6.38%
- Odds against by the river7.01 : 1
- Rule of 4 estimate-0.49 points from the exact chance12%
The count behind 3 outs on the flop
This count leaves you empty-handed 87.51% of the time. A 12.49% chance can still be worth chasing, but only when the bet you face is a small fraction of the pot.
- Unseen cards47
- Your outs3
- Cards that miss44
- Pairs of cards with no out946
- Pairs with at least one out135
- Chance by the river12.49%
- Chance on the next card6.38%
- Odds against hitting7.01 : 1
- Estimate from the rule of 412%
- Error of the rule of 4-0.49 points
Chance by the river = 1 - C(u - k, 2) / C(u, 2), with u = 47 unseen cardsFor 3 outs: 1 - C(44, 2) / C(47, 2) = 1 - 946 / 1,081 = 12.49%Next card = 3 / 47 = 6.38%
Think in pairs. The 47 unseen cards form 1,081 pairs, and only 946 avoid every out. Subtract and you land on 135 helpful pairs, a 12.49% chance.
The quick estimate of 12% and the exact 12.49% nearly agree, separated by 0.49 points.
No adjustment is needed on the flop with 3 outs. The plain rule of 4 gives 4 x 3 = 12%.
Which holding has 3 outs?
One overcard to top pair fits this count exactly, with 3 outs. Memorizing counts like this lets you read the chance straight from the table on this page.
A shared out has a price: 4.07 points for the first, and 12.49 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 | 3 | 12.49% | 0.00 points |
| 1 | 2 | 8.42% | 4.07 points |
| 2 | 1 | 4.26% | 8.23 points |
| 3 | 0 | 0.00% | 12.49 points |
The bets 3 outs can call on the flop
The table prices every bet against 12.49%. Margin is your chance minus the equity a call needs, and no bet size is a call.
Past 16.65% of the pot, the price beats your 12.49%. On a $100 pot that line is $16.65.
| Bet size | Equity a call needs | Margin over it | Verdict |
|---|---|---|---|
| 1/4 pot | 16.67% | -4.18 points | Fold |
| 1/3 pot | 20.00% | -7.51 points | Fold |
| 2/5 pot | 22.22% | -9.73 points | Fold |
| 1/2 pot | 25.00% | -12.51 points | Fold |
| 2/3 pot | 28.57% | -16.08 points | Fold |
| 3/4 pot | 30.00% | -17.51 points | Fold |
| full pot | 33.33% | -20.84 points | Fold |
| 1.25x pot | 35.71% | -23.23 points | Fold |
| 1.5x pot | 37.50% | -25.01 points | Fold |
| 2x pot | 40.00% | -27.51 points | Fold |
| 3x pot | 42.86% | -30.37 points | Fold |
| 4x pot | 44.44% | -31.96 points | Fold |
Nearby counts on both streets
Neighboring counts help you judge a close draw. One out more or less shifts the chance by roughly 3.98 points on this street.
- 3 outs on the turn: 6.52% on the river.
- 2 outs on the flop: 8.42% by the river.
- 4 outs on the flop: 16.47% by the river.
When a bet buys only one card
When more betting is still to come, the fair test is the next card alone. 6.38% against the 25.00% that a half-pot call needs is -18.62 points, which reads Fold.
Whether you count one card or two, 3 outs clear none of the 12 bet sizes.
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 3 outs that is 1 - 946 / 1,081 = 12.49%. The chance on the next card is k / 47, here 6.38%.
The rule of 4 figure is 4 x k, here 12%, and its error is that figure minus the exact chance in percentage points, here -0.49 points.
Two assumptions sit behind the page. First, all 47 unseen cards are equally likely, with no cards taken out for an opponent. Second, each bet is called in full on a $100 pot, so you pay one bet to play for a pot that holds the pot and both bets.
Computed from the closed form by the GeorgeLiPoker math library.
Updated
Questions
What are the odds of hitting with 3 outs on the flop?
With 3 outs on the flop you hit by the river 12.49% of the time, about 12 of every 100 hands, and on the next card 6.38%. The odds against hitting by the river are 7.01 : 1, since 946 of 1,081 pairs of unseen cards hold no out.
What does the rule of 4 say for a count of 3?
The rule of 4 gives 12% for 3 outs against an exact 12.49%, an error of -0.49 points. That is close enough to use at the table.
How many bet sizes can 3 outs on the flop pay for?
At 12.49% by the river, you can call none of the 12 bet sizes. A half-pot bet needs 25.00%, a margin of -12.51 points, so against half the pot the verdict is Fold. The largest bet you can call is 16.65% of the pot, or $16.65 on a $100 pot.
Should I count every out when a rival could be drawing too, with 3 outs?
Take them off the count first. With one shared out the chance by the river falls from 12.49% to 8.42%, and with 3 shared it falls to 0.00%, a loss of 12.49 points. The effective count is then 0 outs.
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