Chance to hit by the river
67.53%
20 outs, two cards to come
- Chance on the next card42.55%
- Odds against by the river0.48 : 1
- Rule of 4 estimate+12.47 points from the exact chance80%
The count behind 20 outs on the flop
Counts this high behave differently. At 67.53%, the equity a call needs is below your chance for most bet sizes, and sometimes for all of them.
- Unseen cards47
- Your outs20
- Cards that miss27
- Pairs of cards with no out351
- Pairs with at least one out730
- Chance by the river67.53%
- Chance on the next card42.55%
- Odds against hitting0.48 : 1
- Estimate from the rule of 480%
- Error of the rule of 4+12.47 points
- Adjusted rule 4k - (k - 8)68%
- Error of the adjusted rule+0.47 points
Chance by the river = 1 - C(u - k, 2) / C(u, 2), with u = 47 unseen cardsFor 20 outs: 1 - C(27, 2) / C(47, 2) = 1 - 351 / 1,081 = 67.53%Next card = 20 / 47 = 42.55%
Count the ways to miss twice: 27 cards miss, and 351 pairs can be drawn from them. Take that from 1,081 pairs in all and 730 are left, so 67.53% of the run-outs help you.
Rounding in your head to 80% would flatter the draw by 12.47 points. With 20 outs, that gap is big enough to flip a close call.
Take off the outs above 8 and the flop rule becomes 68%, an error of +0.47 points. That is a much better stand-in than 80%.
Which holding has 20 outs?
No single common draw has exactly 20 outs. A count like this comes from combining draws and removing the cards that count twice.
Not every out is clean. Here is the chance once shared outs are removed: from 67.53% down to 59.76% when 3 are shared.
| Shared outs | Effective outs | Chance by the river | Chance given up |
|---|---|---|---|
| 0 | 20 | 67.53% | 0.00 points |
| 1 | 19 | 65.03% | 2.50 points |
| 2 | 18 | 62.44% | 5.09 points |
| 3 | 17 | 59.76% | 7.77 points |
The bets 20 outs can call on the flop
Here is the price list. For each bet size it shows the equity a call needs, your margin over it and the verdict, and every verdict is Call.
Your chance of 67.53% is at least one half, so no bet size is too big: any size is a call, because the equity a call needs never reaches 50%.
| Bet size | Equity a call needs | Margin over it | Verdict |
|---|---|---|---|
| 1/4 pot | 16.67% | +50.86 points | Call |
| 1/3 pot | 20.00% | +47.53 points | Call |
| 2/5 pot | 22.22% | +45.31 points | Call |
| 1/2 pot | 25.00% | +42.53 points | Call |
| 2/3 pot | 28.57% | +38.96 points | Call |
| 3/4 pot | 30.00% | +37.53 points | Call |
| full pot | 33.33% | +34.20 points | Call |
| 1.25x pot | 35.71% | +31.82 points | Call |
| 1.5x pot | 37.50% | +30.03 points | Call |
| 2x pot | 40.00% | +27.53 points | Call |
| 3x pot | 42.86% | +24.67 points | Call |
| 4x pot | 44.44% | +23.09 points | Call |
Nearby counts on both streets
Neighboring counts help you judge a close draw. One out more or less shifts the chance by roughly 2.50 points on this street.
- 20 outs on the turn: 43.48% on the river.
- 19 outs on the flop: 65.03% by the river.
When a bet buys only one card
Facing a half-pot bet, compare 25.00% with 42.55% when you will pay again on the turn. A margin of +17.55 points says Call, while the two-card view gave +42.53 points.
Drop to the next-card chance and the number of calls goes from 12 to 10 out of 12.
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 20 outs that is 1 - 351 / 1,081 = 67.53%. The chance on the next card is k / 47, here 42.55%.
The rule of 4 figure is 4 x k, here 80%, and its error is that figure minus the exact chance in percentage points, here +12.47 points. The adjusted rule 4k - (k - 8) gives 68%, an error of +0.47 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
How good is a draw with 20 outs on the flop?
With 20 outs on the flop you hit by the river 67.53% of the time, about 68 of every 100 hands, and on the next card 42.55%. The odds against hitting by the river are 0.48 : 1, since 351 of 1,081 pairs of unseen cards hold no out.
What does the rule of 4 say for a count of 20?
The rule of 4 gives 80% for 20 outs against an exact 67.53%, an error of +12.47 points, so it overstates the draw. The adjusted rule 4k - (k - 8) gives 68%, an error of +0.47 points, and is the closer figure.
Which bets can 20 outs call on the flop?
At 67.53% by the river, you can call all 12 bet sizes. A half-pot bet needs 25.00%, a margin of +42.53 points, so against half the pot the verdict is Call. Any bet size is a call, because 67.53% is above one half.
What is the effective count if some of the 20 outs are shared?
Take them off the count first. With one shared out the chance by the river falls from 67.53% to 65.03%, and with 3 shared it falls to 59.76%, a loss of 7.77 points. The effective count is then 17 outs.
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