Chance to hit on the river
10.87%
5 outs, one card to come
- Odds against8.2 : 1
- Rule of 2 estimate-0.87 points from the exact chance10%
- Unseen cards46
The count behind 5 outs on the turn
A 10.87% chance is the kind of figure that makes bet size the whole question. A small bet can be called, a large one cannot, and the bets section below puts a number on the line.
- Unseen cards46
- Your outs5
- Cards that miss41
- Chance on the river10.87%
- Chance on the next card (the river)10.87%
- Odds against hitting8.2 : 1
- Estimate from the rule of 210%
- Error of the rule of 2-0.87 points
Chance on the river = k / u, with u = 46 unseen cardsFor 5 outs: 5 / 46 = 10.87%
One card is left to come out of 46 unseen. 5 of them are outs and 41 miss, so the chance is 5 divided by 46, which is 10.87%.
Being 0.87 points low, 10% still points the right way, but the exact 10.87% is the number to put against a bet.
Stay with the rule of 2 for this count: 5 outs times 2 is 10%. The adjusted version is for larger flop counts.
Which holding has 5 outs?
There is no stock draw with exactly 5 outs. The number comes from your own count of helpful cards, with overlaps removed.
Count carefully: a card that gives you the draw and gives an opponent a better hand should come off the list. One shared out takes the chance down by 2.17 points, to 8.70%.
| Shared outs | Effective outs | Chance on the river | Chance given up |
|---|---|---|---|
| 0 | 5 | 10.87% | 0.00 points |
| 1 | 4 | 8.70% | 2.17 points |
| 2 | 3 | 6.52% | 4.35 points |
| 3 | 2 | 4.35% | 6.52 points |
The bets 5 outs can call on the turn
A call is right when your chance beats what the bet asks of you. At 10.87%, that holds for none of the 12 bet sizes in the table.
Work the price backward from 10.87% and you find a limit of 13.89% of the pot: $13.89 into a $100 pot.
| Bet size | Equity a call needs | Margin over it | Verdict |
|---|---|---|---|
| 1/4 pot | 16.67% | -5.80 points | Fold |
| 1/3 pot | 20.00% | -9.13 points | Fold |
| 2/5 pot | 22.22% | -11.35 points | Fold |
| 1/2 pot | 25.00% | -14.13 points | Fold |
| 2/3 pot | 28.57% | -17.70 points | Fold |
| 3/4 pot | 30.00% | -19.13 points | Fold |
| full pot | 33.33% | -22.46 points | Fold |
| 1.25x pot | 35.71% | -24.84 points | Fold |
| 1.5x pot | 37.50% | -26.63 points | Fold |
| 2x pot | 40.00% | -29.13 points | Fold |
| 3x pot | 42.86% | -31.99 points | Fold |
| 4x pot | 44.44% | -33.57 points | Fold |
Nearby counts on both streets
Adding or losing one out moves the chance by about 2.17 points on the turn. Here are the counts next to 5, with the other street alongside.
- 5 outs on the flop: 20.35% by the river.
- 4 outs on the turn: 8.70% on the river.
- 6 outs on the turn: 13.04% on the river.
The river is the last card
The turn call buys the last card. 10.87% is the chance, 25.00% is the price of a half-pot bet, and the margin of -14.13 points gives Fold.
How this is computed
The chance to hit on the river from the turn is k / 46, where 46 is the number of unseen cards and k is your effective outs. For 5 outs that is 5 / 46 = 10.87%. The next card is the river, so both chances are the same.
The rule of 2 figure is 2 x k, here 10%, and its error is that figure minus the exact chance in percentage points, here -0.87 points.
The model gives every one of the 46 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 5 outs on the turn?
With 5 outs on the turn you hit on the river 10.87% of the time, about 11 of every 100 hands, because 5 of the 46 unseen cards are outs. The odds against are 8.2 : 1.
Is the rule of 2 close enough for 5 outs?
The rule of 2 gives 10% for 5 outs against an exact 10.87%, an error of -0.87 points. The shortcut sits below the true chance, so the exact figure is better for you.
Are 5 outs on the turn enough to call a half-pot bet?
At 10.87% on the river, you can call none of the 12 bet sizes. A half-pot bet needs 25.00%, a margin of -14.13 points, so against half the pot the verdict is Fold. The largest bet you can call is 13.89% of the pot, or $13.89 on a $100 pot.
What is the effective count if some of the 5 outs are shared?
Take them off the count first. With one shared out the chance on the river falls from 10.87% to 8.70%, and with 3 shared it falls to 4.35%, a loss of 6.52 points. The effective count is then 2 outs.
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