Chance to hit by the river
16.47%
4 outs, two cards to come
- Chance on the next card8.51%
- Odds against by the river5.07 : 1
- Rule of 4 estimate-0.47 points from the exact chance16%
The count behind 4 outs on the flop
Think of 16.47% as a long shot that pays when the price is right. At 5.07 : 1 against, you need the pot to be laying a generous price.
- Unseen cards47
- Your outs4
- Cards that miss43
- Pairs of cards with no out903
- Pairs with at least one out178
- Chance by the river16.47%
- Chance on the next card8.51%
- Odds against hitting5.07 : 1
- Estimate from the rule of 416%
- Error of the rule of 4-0.47 points
Chance by the river = 1 - C(u - k, 2) / C(u, 2), with u = 47 unseen cardsFor 4 outs: 1 - C(43, 2) / C(47, 2) = 1 - 903 / 1,081 = 16.47%Next card = 4 / 47 = 8.51%
The draw fails only when the turn and the river both miss. That covers 903 of the 1,081 pairs, so it lands in the other 178, or 16.47%.
The quick estimate of 16% and the exact 16.47% nearly agree, separated by 0.47 points.
Leave the adjusted rule for bigger flop draws. With 4 outs on the flop, the plain rule of 4 figure is 16%.
Which holding has 4 outs?
One standard source of this count is a gutshot straight draw. Count the cards carefully at the table, because outs that also help a rival do not count in full.
Not every out is clean. Here is the chance once shared outs are removed: from 16.47% down to 4.26% when 3 are shared.
| Shared outs | Effective outs | Chance by the river | Chance given up |
|---|---|---|---|
| 0 | 4 | 16.47% | 0.00 points |
| 1 | 3 | 12.49% | 3.98 points |
| 2 | 2 | 8.42% | 8.05 points |
| 3 | 1 | 4.26% | 12.21 points |
The bets 4 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 no verdict is Call.
Past 24.55% of the pot, the price beats your 16.47%. On a $100 pot that line is $24.55.
| Bet size | Equity a call needs | Margin over it | Verdict |
|---|---|---|---|
| 1/4 pot | 16.67% | -0.20 points | Fold |
| 1/3 pot | 20.00% | -3.53 points | Fold |
| 2/5 pot | 22.22% | -5.76 points | Fold |
| 1/2 pot | 25.00% | -8.53 points | Fold |
| 2/3 pot | 28.57% | -12.11 points | Fold |
| 3/4 pot | 30.00% | -13.53 points | Fold |
| full pot | 33.33% | -16.87 points | Fold |
| 1.25x pot | 35.71% | -19.25 points | Fold |
| 1.5x pot | 37.50% | -21.03 points | Fold |
| 2x pot | 40.00% | -23.53 points | Fold |
| 3x pot | 42.86% | -26.39 points | Fold |
| 4x pot | 44.44% | -27.98 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 3.89 points on this street.
- 4 outs on the turn: 8.70% on the river.
- 3 outs on the flop: 12.49% by the river.
- 5 outs on the flop: 20.35% by the river.
When a bet buys only one card
Price the flop call on one card. With 4 outs, that is 8.51%, and a half-pot bet asks for 25.00%. The margin of -16.49 points gives the verdict Fold.
No row flips when you drop to the next card. The count of calls stays 0 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 4 outs that is 1 - 903 / 1,081 = 16.47%. The chance on the next card is k / 47, here 8.51%.
The rule of 4 figure is 4 x k, here 16%, and its error is that figure minus the exact chance in percentage points, here -0.47 points.
Model notes: every unseen card among the 47 has the same chance of coming, no rival hand is removed, and each bet is matched by one call on a $100 pot, which is the whole amount a hit wins.
Computed from the closed form by the GeorgeLiPoker math library.
Updated
Questions
What is the chance to hit with 4 outs on the flop?
With 4 outs on the flop you hit by the river 16.47% of the time, about 16 of every 100 hands, and on the next card 8.51%. The odds against hitting by the river are 5.07 : 1, since 903 of 1,081 pairs of unseen cards hold no out.
Can I trust the shortcut for 4 outs on the flop?
The rule of 4 gives 16% for 4 outs against an exact 16.47%, an error of -0.47 points. That is close enough to use at the table.
Do 4 outs on the flop justify a call, and against which bets?
At 16.47% by the river, you can call none of the 12 bet sizes. A half-pot bet needs 25.00%, a margin of -8.53 points, so against half the pot the verdict is Fold. The largest bet you can call is 24.55% of the pot, or $24.55 on a $100 pot.
What if some of my 4 outs also help a better hand?
Take them off the count first. With one shared out the chance by the river falls from 16.47% to 12.49%, and with 3 shared it falls to 4.26%, a loss of 12.21 points. The effective count is then 1 out.
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