Chance to hit by the river
38.39%
10 outs, two cards to come
- Chance on the next card21.28%
- Odds against by the river1.6 : 1
- Rule of 4 estimate+1.61 points from the exact chance40%
The count behind 10 outs on the flop
With 38.39% by the river, the draw fails 61.61% of the time and lands the rest. That is enough to call a bet of up to 165.34% of the pot.
- Unseen cards47
- Your outs10
- Cards that miss37
- Pairs of cards with no out666
- Pairs with at least one out415
- Chance by the river38.39%
- Chance on the next card21.28%
- Odds against hitting1.6 : 1
- Estimate from the rule of 440%
- Error of the rule of 4+1.61 points
- Adjusted rule 4k - (k - 8)38%
- Error of the adjusted rule-0.39 points
Chance by the river = 1 - C(u - k, 2) / C(u, 2), with u = 47 unseen cardsFor 10 outs: 1 - C(37, 2) / C(47, 2) = 1 - 666 / 1,081 = 38.39%Next card = 10 / 47 = 21.28%
Think in pairs. The 47 unseen cards form 1,081 pairs, and only 666 avoid every out. Subtract and you land on 415 helpful pairs, a 38.39% chance.
With 10 outs the rule of 4 stays near the truth: 40% estimated, 38.39% exact, 1.61 points apart.
Above 8 outs the adjusted rule applies: 4 x 10 - 2 = 38%. Against 38.39%, it is off by 0.39 points.
Which holding has 10 outs?
Look at the board and your hand, count each card that helps once, and reach 10. No single named draw has this count by itself.
Use the effective count, not the raw one, when a rival could be drawing too. Losing 3 outs to overlap moves 38.39% down to 27.84%.
| Shared outs | Effective outs | Chance by the river | Chance given up |
|---|---|---|---|
| 0 | 10 | 38.39% | 0.00 points |
| 1 | 9 | 34.97% | 3.42 points |
| 2 | 8 | 31.45% | 6.94 points |
| 3 | 7 | 27.84% | 10.55 points |
The bets 10 outs can call on the flop
Compare the columns: required equity on the left, margin in the middle, verdict at the end. With 10 outs you can call 9 of the 12 bet sizes.
Past 165.34% of the pot, the price beats your 38.39%. On a $100 pot that line is $165.34.
| Bet size | Equity a call needs | Margin over it | Verdict |
|---|---|---|---|
| 1/4 pot | 16.67% | +21.72 points | Call |
| 1/3 pot | 20.00% | +18.39 points | Call |
| 2/5 pot | 22.22% | +16.17 points | Call |
| 1/2 pot | 25.00% | +13.39 points | Call |
| 2/3 pot | 28.57% | +9.82 points | Call |
| 3/4 pot | 30.00% | +8.39 points | Call |
| full pot | 33.33% | +5.06 points | Call |
| 1.25x pot | 35.71% | +2.68 points | Call |
| 1.5x pot | 37.50% | +0.89 points | Call |
| 2x pot | 40.00% | -1.61 points | Fold |
| 3x pot | 42.86% | -4.47 points | Fold |
| 4x pot | 44.44% | -6.05 points | Fold |
Nearby counts on both streets
Not sure whether to count one more out? It changes the chance by about 3.33 points. The table shows both streets for the counts around 10.
- 10 outs on the turn: 21.74% on the river.
- 9 outs on the flop: 34.97% by the river.
- 11 outs on the flop: 41.72% by the river.
When a bet buys only one card
Facing a half-pot bet, compare 25.00% with 21.28% when you will pay again on the turn. A margin of -3.72 points says Fold, while the two-card view gave +13.39 points.
Counting the next card alone, you can call 2 of the 12 bet sizes, compared with 9 of the 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 10 outs that is 1 - 666 / 1,081 = 38.39%. The chance on the next card is k / 47, here 21.28%.
The rule of 4 figure is 4 x k, here 40%, and its error is that figure minus the exact chance in percentage points, here +1.61 points. The adjusted rule 4k - (k - 8) gives 38%, an error of -0.39 points.
The count covers every card you cannot see. All 47 unseen cards stay in it, the call is paid in full, and the pot is $100 before the bet, so the verdicts are a plain price check.
Computed from the closed form by the GeorgeLiPoker math library.
Updated
Questions
How good is a draw with 10 outs on the flop?
With 10 outs on the flop you hit by the river 38.39% of the time, about 38 of every 100 hands, and on the next card 21.28%. The odds against hitting by the river are 1.6 : 1, since 666 of 1,081 pairs of unseen cards hold no out.
What is the error of the rule of 4 at 10 outs?
The rule of 4 gives 40% for 10 outs against an exact 38.39%, an error of +1.61 points, so it runs a little high on the draw. The adjusted rule 4k - (k - 8) gives 38%, an error of -0.39 points, and is the closer figure.
Are 10 outs on the flop enough to call a half-pot bet?
At 38.39% by the river, you can call 9 of the 12 bet sizes. A half-pot bet needs 25.00%, a margin of +13.39 points, so against half the pot the verdict is Call. The largest bet you can call is 165.34% of the pot, or $165.34 on a $100 pot.
What is the effective count if some of the 10 outs are shared?
Take them off the count first. With one shared out the chance by the river falls from 38.39% to 34.97%, and with 3 shared it falls to 27.84%, a loss of 10.55 points. The effective count is then 7 outs.
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