Chance to hit by the river
27.84%
7 outs, two cards to come
- Chance on the next card14.89%
- Odds against by the river2.59 : 1
- Rule of 4 estimate+0.16 points from the exact chance28%
The count behind 7 outs on the flop
Between one chance in five and two in five, 27.84% is a draw worth playing when the bet is not oversized.
- Unseen cards47
- Your outs7
- Cards that miss40
- Pairs of cards with no out780
- Pairs with at least one out301
- Chance by the river27.84%
- Chance on the next card14.89%
- Odds against hitting2.59 : 1
- Estimate from the rule of 428%
- Error of the rule of 4+0.16 points
Chance by the river = 1 - C(u - k, 2) / C(u, 2), with u = 47 unseen cardsFor 7 outs: 1 - C(40, 2) / C(47, 2) = 1 - 780 / 1,081 = 27.84%Next card = 7 / 47 = 14.89%
Of the 1,081 pairs that two cards can make from the 47 unseen, 780 contain only misses. The other 301 hold at least one out, which is 27.84% of the pairs.
The quick estimate of 28% and the exact 27.84% nearly agree, separated by 0.16 points.
The adjusted rule only applies from the flop above 8 outs, so 7 outs on the flop use the plain rule of 4: 4 x 7 = 28%.
Which holding has 7 outs?
This count has no one textbook draw behind it. It is the total you get after adding up several draws and checking for overlap.
An out that also completes a better hand for a rival is a poor out. With one shared out among your 7 outs, the chance by the river falls from 27.84% to 24.14%, a loss of 3.70 points.
| Shared outs | Effective outs | Chance by the river | Chance given up |
|---|---|---|---|
| 0 | 7 | 27.84% | 0.00 points |
| 1 | 6 | 24.14% | 3.70 points |
| 2 | 5 | 20.35% | 7.49 points |
| 3 | 4 | 16.47% | 11.38 points |
The bets 7 outs can call on the flop
Draw the line at 27.84%: any bet that needs less equity than that is a call. Here that covers 4 of the 12 bet sizes.
Work the price backward from 27.84% and you find a limit of 62.84% of the pot: $62.84 into a $100 pot.
| Bet size | Equity a call needs | Margin over it | Verdict |
|---|---|---|---|
| 1/4 pot | 16.67% | +11.18 points | Call |
| 1/3 pot | 20.00% | +7.84 points | Call |
| 2/5 pot | 22.22% | +5.62 points | Call |
| 1/2 pot | 25.00% | +2.84 points | Call |
| 2/3 pot | 28.57% | -0.73 points | Fold |
| 3/4 pot | 30.00% | -2.16 points | Fold |
| full pot | 33.33% | -5.49 points | Fold |
| 1.25x pot | 35.71% | -7.87 points | Fold |
| 1.5x pot | 37.50% | -9.66 points | Fold |
| 2x pot | 40.00% | -12.16 points | Fold |
| 3x pot | 42.86% | -15.01 points | Fold |
| 4x pot | 44.44% | -16.60 points | Fold |
Nearby counts on both streets
Counts near 7 sit side by side below, on the flop and on the other street. One out is worth around 3.61 points on this page.
- 7 outs on the turn: 15.22% on the river.
- 6 outs on the flop: 24.14% by the river.
- 8 outs on the flop: 31.45% by the river.
When a bet buys only one card
Think about what the bet buys. If you must call again on the turn, only 14.89% counts for the flop call. A half-pot price of 25.00% leaves -10.11 points, which is a fold.
Drop to the next-card chance and the number of calls goes from 4 to 0 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 7 outs that is 1 - 780 / 1,081 = 27.84%. The chance on the next card is k / 47, here 14.89%.
The rule of 4 figure is 4 x k, here 28%, and its error is that figure minus the exact chance in percentage points, here +0.16 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
How often do 7 outs hit on the flop?
With 7 outs on the flop you hit by the river 27.84% of the time, about 28 of every 100 hands, and on the next card 14.89%. The odds against hitting by the river are 2.59 : 1, since 780 of 1,081 pairs of unseen cards hold no out.
How does the quick rule compare with the exact chance for 7 outs?
The rule of 4 gives 28% for 7 outs against an exact 27.84%, an error of +0.16 points. That is close enough to use at the table.
How big a bet can I call with 7 outs on the flop?
At 27.84% by the river, you can call 4 of the 12 bet sizes. A half-pot bet needs 25.00%, a margin of +2.84 points, so against half the pot the verdict is Call. The largest bet you can call is 62.84% of the pot, or $62.84 on a $100 pot.
How many points do I lose if outs are shared with a better hand?
Take them off the count first. With one shared out the chance by the river falls from 27.84% to 24.14%, and with 3 shared it falls to 16.47%, a loss of 11.38 points. The effective count is then 4 outs.
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