Chance to hit by the river
41.72%
11 outs, two cards to come
- Chance on the next card23.40%
- Odds against by the river1.4 : 1
- Rule of 4 estimate+2.28 points from the exact chance44%
The count behind 11 outs on the flop
At 41.72% the draw is a big one. The odds against are only 1.4 : 1.
- Unseen cards47
- Your outs11
- Cards that miss36
- Pairs of cards with no out630
- Pairs with at least one out451
- Chance by the river41.72%
- Chance on the next card23.40%
- Odds against hitting1.4 : 1
- Estimate from the rule of 444%
- Error of the rule of 4+2.28 points
- Adjusted rule 4k - (k - 8)41%
- Error of the adjusted rule-0.72 points
Chance by the river = 1 - C(u - k, 2) / C(u, 2), with u = 47 unseen cardsFor 11 outs: 1 - C(36, 2) / C(47, 2) = 1 - 630 / 1,081 = 41.72%Next card = 11 / 47 = 23.40%
Here is the arithmetic for 11 outs. Both cards must miss for the draw to fail: that is 630 pairs out of 1,081. The remaining 451 pairs give you 41.72%.
Overstating by 2.28 points, the rule of 4 is too rosy for 11 outs. Take the exact 41.72% when you compare with a bet.
Above 8 outs the adjusted rule applies: 4 x 11 - 3 = 41%. Against 41.72%, it is off by 0.72 points.
Which holding has 11 outs?
A count of 11 is a combination, not a textbook draw. Add up the helpful cards from each part of your hand and take away any that repeat.
Shared outs are a quiet leak. From 41.72%, a single shared out leaves 38.39%, and 3 of them leave 31.45%.
| Shared outs | Effective outs | Chance by the river | Chance given up |
|---|---|---|---|
| 0 | 11 | 41.72% | 0.00 points |
| 1 | 10 | 38.39% | 3.33 points |
| 2 | 9 | 34.97% | 6.75 points |
| 3 | 8 | 31.45% | 10.27 points |
The bets 11 outs can call on the flop
Bet sizes run from 1/4 pot to 4x pot. Against 41.72% by the river, 10 of the 12 bet sizes are calls.
Past 251.96% of the pot, the price beats your 41.72%. On a $100 pot that line is $251.96.
| Bet size | Equity a call needs | Margin over it | Verdict |
|---|---|---|---|
| 1/4 pot | 16.67% | +25.05 points | Call |
| 1/3 pot | 20.00% | +21.72 points | Call |
| 2/5 pot | 22.22% | +19.50 points | Call |
| 1/2 pot | 25.00% | +16.72 points | Call |
| 2/3 pot | 28.57% | +13.15 points | Call |
| 3/4 pot | 30.00% | +11.72 points | Call |
| full pot | 33.33% | +8.39 points | Call |
| 1.25x pot | 35.71% | +6.01 points | Call |
| 1.5x pot | 37.50% | +4.22 points | Call |
| 2x pot | 40.00% | +1.72 points | Call |
| 3x pot | 42.86% | -1.14 points | Fold |
| 4x pot | 44.44% | -2.72 points | Fold |
Nearby counts on both streets
Adding or losing one out moves the chance by about 3.24 points on the flop. Here are the counts next to 11, with the other street alongside.
- 11 outs on the turn: 23.91% on the river.
- 10 outs on the flop: 38.39% by the river.
- 12 outs on the flop: 44.96% 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. 23.40% against the 25.00% that a half-pot call needs is -1.60 points, which reads Fold.
On one card, 11 outs clear 3 of the 12 bet sizes. On two cards they clear 10 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 11 outs that is 1 - 630 / 1,081 = 41.72%. The chance on the next card is k / 47, here 23.40%.
The rule of 4 figure is 4 x k, here 44%, and its error is that figure minus the exact chance in percentage points, here +2.28 points. The adjusted rule 4k - (k - 8) gives 41%, an error of -0.72 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
Out of 100 hands, how many hit with 11 outs on the flop?
With 11 outs on the flop you hit by the river 41.72% of the time, about 42 of every 100 hands, and on the next card 23.40%. The odds against hitting by the river are 1.4 : 1, since 630 of 1,081 pairs of unseen cards hold no out.
What is the error of the rule of 4 at 11 outs?
The rule of 4 gives 44% for 11 outs against an exact 41.72%, an error of +2.28 points, so it overstates the draw. The adjusted rule 4k - (k - 8) gives 41%, an error of -0.72 points, and is the closer figure.
What price can a draw with 11 outs on the flop afford?
At 41.72% by the river, you can call 10 of the 12 bet sizes. A half-pot bet needs 25.00%, a margin of +16.72 points, so against half the pot the verdict is Call. The largest bet you can call is 251.96% of the pot, or $251.96 on a $100 pot.
What if some of my 11 outs also help a better hand?
Take them off the count first. With one shared out the chance by the river falls from 41.72% to 38.39%, and with 3 shared it falls to 31.45%, a loss of 10.27 points. The effective count is then 8 outs.
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