Chance to hit by the river
48.10%
13 outs, two cards to come
- Chance on the next card27.66%
- Odds against by the river1.08 : 1
- Rule of 4 estimate+3.90 points from the exact chance52%
The count behind 13 outs on the flop
A 48.10% chance is the territory of big draws. The miss is 51.90%, so even a bet the size of the pot can be a call.
- Unseen cards47
- Your outs13
- Cards that miss34
- Pairs of cards with no out561
- Pairs with at least one out520
- Chance by the river48.10%
- Chance on the next card27.66%
- Odds against hitting1.08 : 1
- Estimate from the rule of 452%
- Error of the rule of 4+3.90 points
- Adjusted rule 4k - (k - 8)47%
- Error of the adjusted rule-1.10 points
Chance by the river = 1 - C(u - k, 2) / C(u, 2), with u = 47 unseen cardsFor 13 outs: 1 - C(34, 2) / C(47, 2) = 1 - 561 / 1,081 = 48.10%Next card = 13 / 47 = 27.66%
Here is the arithmetic for 13 outs. Both cards must miss for the draw to fail: that is 561 pairs out of 1,081. The remaining 520 pairs give you 48.10%.
52% from the rule of 4 sits 3.90 points over the true 48.10%. The larger the count of outs on the flop, the further the shortcut drifts.
Subtracting the 5 outs beyond 8 gives 47%. That lands 1.10 points from 48.10%, so it is the shortcut to prefer here.
Which holding has 13 outs?
There is no stock draw with exactly 13 outs. The number comes from your own count of helpful cards, with overlaps removed.
A shared out has a price: 3.15 points for the first, and 9.71 points in all with 3 shared. Subtract any shared outs before you compare with a bet.
| Shared outs | Effective outs | Chance by the river | Chance given up |
|---|---|---|---|
| 0 | 13 | 48.10% | 0.00 points |
| 1 | 12 | 44.96% | 3.15 points |
| 2 | 11 | 41.72% | 6.38 points |
| 3 | 10 | 38.39% | 9.71 points |
The bets 13 outs can call on the flop
Draw the line at 48.10%: any bet that needs less equity than that is a call. Here that covers all 12 bet sizes.
Anything above $1,268.29 on a $100 pot, which is 1,268.29% of the pot, asks for more equity than this draw holds.
| Bet size | Equity a call needs | Margin over it | Verdict |
|---|---|---|---|
| 1/4 pot | 16.67% | +31.44 points | Call |
| 1/3 pot | 20.00% | +28.10 points | Call |
| 2/5 pot | 22.22% | +25.88 points | Call |
| 1/2 pot | 25.00% | +23.10 points | Call |
| 2/3 pot | 28.57% | +19.53 points | Call |
| 3/4 pot | 30.00% | +18.10 points | Call |
| full pot | 33.33% | +14.77 points | Call |
| 1.25x pot | 35.71% | +12.39 points | Call |
| 1.5x pot | 37.50% | +10.60 points | Call |
| 2x pot | 40.00% | +8.10 points | Call |
| 3x pot | 42.86% | +5.25 points | Call |
| 4x pot | 44.44% | +3.66 points | Call |
Nearby counts on both streets
Adding or losing one out moves the chance by about 3.05 points on the flop. Here are the counts next to 13, with the other street alongside.
- 13 outs on the turn: 28.26% on the river.
- 12 outs on the flop: 44.96% by the river.
- 14 outs on the flop: 51.16% by the river.
When a bet buys only one card
Check the one-card case before you call a flop bet. 27.66% against 25.00% is +2.66 points for 13 outs, and that is Call against a half-pot bet.
The one-card view cuts the bet sizes you can call to 4, from 12 in the table above.
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 13 outs that is 1 - 561 / 1,081 = 48.10%. The chance on the next card is k / 47, here 27.66%.
The rule of 4 figure is 4 x k, here 52%, and its error is that figure minus the exact chance in percentage points, here +3.90 points. The adjusted rule 4k - (k - 8) gives 47%, an error of -1.10 points.
The model gives every one of the 47 unseen cards the same chance and removes nothing for what a rival holds. It measures hitting an out, not winning the hand. Bets are priced on a $100 pot, and a call adds one bet to the pot.
Computed from the closed form by the GeorgeLiPoker math library.
Updated
Questions
How should I read the hit chance for 13 outs on the flop?
With 13 outs on the flop you hit by the river 48.10% of the time, about 48 of every 100 hands, and on the next card 27.66%. The odds against hitting by the river are 1.08 : 1, since 561 of 1,081 pairs of unseen cards hold no out.
How accurate is the rule of 4 for 13 outs?
The rule of 4 gives 52% for 13 outs against an exact 48.10%, an error of +3.90 points, so it overstates the draw. The adjusted rule 4k - (k - 8) gives 47%, an error of -1.10 points, and is the closer figure.
Are 13 outs on the flop enough to call a half-pot bet?
At 48.10% by the river, you can call all 12 bet sizes. A half-pot bet needs 25.00%, a margin of +23.10 points, so against half the pot the verdict is Call. The largest bet you can call is 1,268.29% of the pot, or $1,268.29 on a $100 pot.
Should I count every out when a rival could be drawing too, with 13 outs?
Take them off the count first. With one shared out the chance by the river falls from 48.10% to 44.96%, and with 3 shared it falls to 38.39%, a loss of 9.71 points. The effective count is then 10 outs.
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