Chance to hit by the river
34.97%
9 outs, two cards to come
- Chance on the next card19.15%
- Odds against by the river1.86 : 1
- Rule of 4 estimate+1.03 points from the exact chance36%
The count behind 9 outs on the flop
Between one chance in five and two in five, 34.97% is a draw worth playing when the bet is not oversized.
- Unseen cards47
- Your outs9
- Cards that miss38
- Pairs of cards with no out703
- Pairs with at least one out378
- Chance by the river34.97%
- Chance on the next card19.15%
- Odds against hitting1.86 : 1
- Estimate from the rule of 436%
- Error of the rule of 4+1.03 points
- Adjusted rule 4k - (k - 8)35%
- Error of the adjusted rule+0.03 points
Chance by the river = 1 - C(u - k, 2) / C(u, 2), with u = 47 unseen cardsFor 9 outs: 1 - C(38, 2) / C(47, 2) = 1 - 703 / 1,081 = 34.97%Next card = 9 / 47 = 19.15%
The draw fails only when the turn and the river both miss. That covers 703 of the 1,081 pairs, so it lands in the other 378, or 34.97%.
The rule of 4 gives 36% against an exact 34.97%, a gap of +1.03 points. That is close enough to use at the table.
The adjusted rule 4k - (k - 8) gives 35% for 9 outs, 0.03 points from the exact 34.97%. It trims the shortcut back toward the truth.
Which holding has 9 outs?
For a flush draw, the count is exactly 9. The chance you see here is the right one to use, as long as none of those cards help an opponent too.
Use the effective count, not the raw one, when a rival could be drawing too. Losing 3 outs to overlap moves 34.97% down to 24.14%.
| Shared outs | Effective outs | Chance by the river | Chance given up |
|---|---|---|---|
| 0 | 9 | 34.97% | 0.00 points |
| 1 | 8 | 31.45% | 3.52 points |
| 2 | 7 | 27.84% | 7.12 points |
| 3 | 6 | 24.14% | 10.82 points |
The bets 9 outs can call on the flop
Compare the columns: required equity on the left, margin in the middle, verdict at the end. With 9 outs you can call 7 of the 12 bet sizes.
The line between call and fold is a bet of 116.31% of the pot, or $116.31 on a $100 pot. Stay at or under it and the draw pays for itself.
| Bet size | Equity a call needs | Margin over it | Verdict |
|---|---|---|---|
| 1/4 pot | 16.67% | +18.30 points | Call |
| 1/3 pot | 20.00% | +14.97 points | Call |
| 2/5 pot | 22.22% | +12.75 points | Call |
| 1/2 pot | 25.00% | +9.97 points | Call |
| 2/3 pot | 28.57% | +6.40 points | Call |
| 3/4 pot | 30.00% | +4.97 points | Call |
| full pot | 33.33% | +1.63 points | Call |
| 1.25x pot | 35.71% | -0.75 points | Fold |
| 1.5x pot | 37.50% | -2.53 points | Fold |
| 2x pot | 40.00% | -5.03 points | Fold |
| 3x pot | 42.86% | -7.89 points | Fold |
| 4x pot | 44.44% | -9.48 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 3.42 points on this street.
- 9 outs on the turn: 19.57% on the river.
- 8 outs on the flop: 31.45% by the river.
- 10 outs on the flop: 38.39% by the river.
When a bet buys only one card
The by-the-river figure of 34.97% assumes you see both cards. If a turn bet follows, use 19.15%: against a half-pot price of 25.00%, that margin is -5.85 points and the verdict is Fold.
1 of 12 sizes survives the one-card test, where the two-card test passed 7.
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 9 outs that is 1 - 703 / 1,081 = 34.97%. The chance on the next card is k / 47, here 19.15%.
The rule of 4 figure is 4 x k, here 36%, and its error is that figure minus the exact chance in percentage points, here +1.03 points. The adjusted rule 4k - (k - 8) gives 35%, an error of +0.03 points.
Seen as a pure card count, the 47 unseen cards are all equally likely and an opponent's hand removes none of them. Prices use a $100 pot and a call equal to the bet, so the equity needed is the bet over the pot plus two bets.
Computed from the closed form by the GeorgeLiPoker math library.
Updated
Questions
What is the chance to hit with 9 outs on the flop?
With 9 outs on the flop you hit by the river 34.97% of the time, about 35 of every 100 hands, and on the next card 19.15%. The odds against hitting by the river are 1.86 : 1, since 703 of 1,081 pairs of unseen cards hold no out.
What is the error of the rule of 4 at 9 outs?
The rule of 4 gives 36% for 9 outs against an exact 34.97%, an error of +1.03 points, so it runs a little high on the draw. The adjusted rule 4k - (k - 8) gives 35%, an error of +0.03 points, and is the closer figure.
How many bet sizes can 9 outs on the flop pay for?
At 34.97% by the river, you can call 7 of the 12 bet sizes. A half-pot bet needs 25.00%, a margin of +9.97 points, so against half the pot the verdict is Call. The largest bet you can call is 116.31% of the pot, or $116.31 on a $100 pot.
What if some of my 9 outs also help a better hand?
Take them off the count first. With one shared out the chance by the river falls from 34.97% to 31.45%, and with 3 shared it falls to 24.14%, a loss of 10.82 points. The effective count is then 6 outs.
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