Chance to hit by the river
31.45%
8 outs, two cards to come
- Chance on the next card17.02%
- Odds against by the river2.18 : 1
- Rule of 4 estimate+0.55 points from the exact chance32%
The count behind 8 outs on the flop
A 31.45% chance gives the draw real weight. You will miss 68.55% of the time, yet a bet of up to 84.79% of the pot still costs less than your chance.
- Unseen cards47
- Your outs8
- Cards that miss39
- Pairs of cards with no out741
- Pairs with at least one out340
- Chance by the river31.45%
- Chance on the next card17.02%
- Odds against hitting2.18 : 1
- Estimate from the rule of 432%
- Error of the rule of 4+0.55 points
Chance by the river = 1 - C(u - k, 2) / C(u, 2), with u = 47 unseen cardsFor 8 outs: 1 - C(39, 2) / C(47, 2) = 1 - 741 / 1,081 = 31.45%Next card = 8 / 47 = 17.02%
39 of the 47 unseen cards are blanks. Two blanks can be dealt in 741 ways, against 1,081 ways to deal any two cards, which leaves 31.45% for the draw.
A gap of 0.55 points between 32% and 31.45% is small. The shortcut does its job with this count.
The correction for big counts belongs to the flop above 8 outs. Here the plain figure, 4 x 8 = 32%, is the one to use.
Which holding has 8 outs?
An open-ended straight draw fits this count exactly, with 8 outs. Memorizing counts like this lets you read the chance straight from the table on this page.
Not every out is clean. Here is the chance once shared outs are removed: from 31.45% down to 20.35% when 3 are shared.
| Shared outs | Effective outs | Chance by the river | Chance given up |
|---|---|---|---|
| 0 | 8 | 31.45% | 0.00 points |
| 1 | 7 | 27.84% | 3.61 points |
| 2 | 6 | 24.14% | 7.31 points |
| 3 | 5 | 20.35% | 11.10 points |
The bets 8 outs can call on the flop
Read the table from the top: small bets need little equity, and the requirement climbs with the bet. Your 31.45% clears 6 of the 12 bet sizes.
The biggest bet you can call is 84.79% of the pot, which is $84.79 on a $100 pot.
| Bet size | Equity a call needs | Margin over it | Verdict |
|---|---|---|---|
| 1/4 pot | 16.67% | +14.79 points | Call |
| 1/3 pot | 20.00% | +11.45 points | Call |
| 2/5 pot | 22.22% | +9.23 points | Call |
| 1/2 pot | 25.00% | +6.45 points | Call |
| 2/3 pot | 28.57% | +2.88 points | Call |
| 3/4 pot | 30.00% | +1.45 points | Call |
| full pot | 33.33% | -1.88 points | Fold |
| 1.25x pot | 35.71% | -4.26 points | Fold |
| 1.5x pot | 37.50% | -6.05 points | Fold |
| 2x pot | 40.00% | -8.55 points | Fold |
| 3x pot | 42.86% | -11.40 points | Fold |
| 4x pot | 44.44% | -12.99 points | Fold |
Nearby counts on both streets
Compare 8 outs with the counts on either side. Each step is worth near 3.52 points on the flop, and the other street is shown next to it.
- 8 outs on the turn: 17.39% on the river.
- 7 outs on the flop: 27.84% by the river.
- 9 outs on the flop: 34.97% by the river.
When a bet buys only one card
The by-the-river figure of 31.45% assumes you see both cards. If a turn bet follows, use 17.02%: against a half-pot price of 25.00%, that margin is -7.98 points and the verdict is Fold.
1 of 12 sizes survives the one-card test, where the two-card test passed 6.
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 8 outs that is 1 - 741 / 1,081 = 31.45%. The chance on the next card is k / 47, here 17.02%.
The rule of 4 figure is 4 x k, here 32%, and its error is that figure minus the exact chance in percentage points, here +0.55 points.
Assumptions: every unseen card is equally likely, and no card is removed for an opponent's hand, so this is the chance to hit your outs, not your equity against a named hand. A call is paid in full, so the pot after it holds the pot plus two bets. Here the pot is $100.
Computed from the closed form by the GeorgeLiPoker math library.
Updated
Questions
What percent chance does a count of 8 give you on the flop?
With 8 outs on the flop you hit by the river 31.45% of the time, about 31 of every 100 hands, and on the next card 17.02%. The odds against hitting by the river are 2.18 : 1, since 741 of 1,081 pairs of unseen cards hold no out.
Can I trust the shortcut for 8 outs on the flop?
The rule of 4 gives 32% for 8 outs against an exact 31.45%, an error of +0.55 points. That is close enough to use at the table.
What price can a draw with 8 outs on the flop afford?
At 31.45% by the river, you can call 6 of the 12 bet sizes. A half-pot bet needs 25.00%, a margin of +6.45 points, so against half the pot the verdict is Call. The largest bet you can call is 84.79% of the pot, or $84.79 on a $100 pot.
What happens to the chance when 8 outs are not all clean?
Take them off the count first. With one shared out the chance by the river falls from 31.45% to 27.84%, and with 3 shared it falls to 20.35%, a loss of 11.10 points. The effective count is then 5 outs.
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