Chance to hit on the river
6.52%
3 outs, one card to come
- Odds against14.33 : 1
- Rule of 2 estimate-0.52 points from the exact chance6%
- Unseen cards46
The count behind 3 outs on the turn
Under one chance in five is thin. 6.52% on the river means the miss is the usual outcome, and the price has to reflect that.
- Unseen cards46
- Your outs3
- Cards that miss43
- Chance on the river6.52%
- Chance on the next card (the river)6.52%
- Odds against hitting14.33 : 1
- Estimate from the rule of 26%
- Error of the rule of 2-0.52 points
Chance on the river = k / u, with u = 46 unseen cardsFor 3 outs: 3 / 46 = 6.52%
43 of the 46 unseen cards miss. A single deal picks one of 46 equally likely cards, which lands on an out 3 times in 46, or 6.52%.
Being 0.52 points low, 6% still points the right way, but the exact 6.52% is the number to put against a bet.
The adjusted rule only applies from the flop above 8 outs, so 3 outs on the turn use the plain rule of 2: 2 x 3 = 6%.
Which holding has 3 outs?
Common holding with this count: one overcard to top pair. This page tells you how often it gets there.
Not every out is clean. Here is the chance once shared outs are removed: from 6.52% down to 0.00% when 3 are shared.
| Shared outs | Effective outs | Chance on the river | Chance given up |
|---|---|---|---|
| 0 | 3 | 6.52% | 0.00 points |
| 1 | 2 | 4.35% | 2.17 points |
| 2 | 1 | 2.17% | 4.35 points |
| 3 | 0 | 0.00% | 6.52 points |
The bets 3 outs can call on the turn
Compare the columns: required equity on the left, margin in the middle, verdict at the end. With 3 outs you can call none of the 12 bet sizes.
Stated as a share of the pot, the largest bet you can pay is 7.50%. On a $100 pot you can call up to $7.50.
| Bet size | Equity a call needs | Margin over it | Verdict |
|---|---|---|---|
| 1/4 pot | 16.67% | -10.14 points | Fold |
| 1/3 pot | 20.00% | -13.48 points | Fold |
| 2/5 pot | 22.22% | -15.70 points | Fold |
| 1/2 pot | 25.00% | -18.48 points | Fold |
| 2/3 pot | 28.57% | -22.05 points | Fold |
| 3/4 pot | 30.00% | -23.48 points | Fold |
| full pot | 33.33% | -26.81 points | Fold |
| 1.25x pot | 35.71% | -29.19 points | Fold |
| 1.5x pot | 37.50% | -30.98 points | Fold |
| 2x pot | 40.00% | -33.48 points | Fold |
| 3x pot | 42.86% | -36.34 points | Fold |
| 4x pot | 44.44% | -37.92 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 2.17 points on this street.
- 3 outs on the flop: 12.49% by the river.
- 2 outs on the turn: 4.35% on the river.
- 4 outs on the turn: 8.70% on the river.
The river is the last card
You do not need a second view on the turn. The 6.52% chance is final, and against a half-pot price of 25.00% it leaves -18.48 points: 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 3 outs that is 3 / 46 = 6.52%. The next card is the river, so both chances are the same.
The rule of 2 figure is 2 x k, here 6%, and its error is that figure minus the exact chance in percentage points, here -0.52 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 often do 3 outs hit on the turn?
With 3 outs on the turn you hit on the river 6.52% of the time, about 7 of every 100 hands, because 3 of the 46 unseen cards are outs. The odds against are 14.33 : 1.
What does the rule of 2 say for a count of 3?
The rule of 2 gives 6% for 3 outs against an exact 6.52%, an error of -0.52 points. The shortcut sits below the true chance, so the exact figure is better for you.
How big a bet can I call with 3 outs on the turn?
At 6.52% on the river, you can call none of the 12 bet sizes. A half-pot bet needs 25.00%, a margin of -18.48 points, so against half the pot the verdict is Fold. The largest bet you can call is 7.50% of the pot, or $7.50 on a $100 pot.
Does every one of the 3 outs count in full on the turn?
Take them off the count first. With one shared out the chance on the river falls from 6.52% to 4.35%, and with 3 shared it falls to 0.00%, a loss of 6.52 points. The effective count is then 0 outs.
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