Chance to hit by the river
4.26%
1 out, two cards to come
- Chance on the next card2.13%
- Odds against by the river22.5 : 1
- Rule of 4 estimate-0.26 points from the exact chance4%
The count behind 1 out on the flop
4.26% is a modest chance. Over many hands the draw lands about 4 times in 100, and that is rarely enough for a large bet.
- Unseen cards47
- Your outs1
- Cards that miss46
- Pairs of cards with no out1,035
- Pairs with at least one out46
- Chance by the river4.26%
- Chance on the next card2.13%
- Odds against hitting22.5 : 1
- Estimate from the rule of 44%
- Error of the rule of 4-0.26 points
Chance by the river = 1 - C(u - k, 2) / C(u, 2), with u = 47 unseen cardsFor 1 out: 1 - C(46, 2) / C(47, 2) = 1 - 1,035 / 1,081 = 4.26%Next card = 1 / 47 = 2.13%
Here is the arithmetic for 1 out. Both cards must miss for the draw to fail: that is 1,035 pairs out of 1,081. The remaining 46 pairs give you 4.26%.
The rule of 4 gives 4% against an exact 4.26%, a gap of -0.26 points. That is close enough to use at the table.
No adjustment is needed on the flop with 1 out. The plain rule of 4 gives 4 x 1 = 4%.
Which holding has 1 out?
A count of 1 means a single card helps. It comes from your own count of helpful cards, not from a standard draw.
Not every out is clean. Here is the chance once shared outs are removed: from 4.26% down to 0.00% when it is shared.
| Shared outs | Effective outs | Chance by the river | Chance given up |
|---|---|---|---|
| 0 | 1 | 4.26% | 0.00 points |
| 1 | 0 | 0.00% | 4.26 points |
The bets 1 out 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.
The line between call and fold is a bet of 4.65% of the pot, or $4.65 on a $100 pot. Stay at or under it and the draw pays for itself.
| Bet size | Equity a call needs | Margin over it | Verdict |
|---|---|---|---|
| 1/4 pot | 16.67% | -12.41 points | Fold |
| 1/3 pot | 20.00% | -15.74 points | Fold |
| 2/5 pot | 22.22% | -17.97 points | Fold |
| 1/2 pot | 25.00% | -20.74 points | Fold |
| 2/3 pot | 28.57% | -24.32 points | Fold |
| 3/4 pot | 30.00% | -25.74 points | Fold |
| full pot | 33.33% | -29.08 points | Fold |
| 1.25x pot | 35.71% | -31.46 points | Fold |
| 1.5x pot | 37.50% | -33.24 points | Fold |
| 2x pot | 40.00% | -35.74 points | Fold |
| 3x pot | 42.86% | -38.60 points | Fold |
| 4x pot | 44.44% | -40.19 points | Fold |
Nearby counts on both streets
Compare 1 out with the counts on either side. Each step is worth near 4.16 points on the flop, and the other street is shown next to it.
- 1 out on the turn: 2.17% on the river.
- 2 outs on the flop: 8.42% 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 2.13% counts for the flop call. A half-pot price of 25.00% leaves -22.87 points, which is a fold.
The verdicts hold up on one card: no bet size is a call, matching the two-card count.
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 1 out that is 1 - 1,035 / 1,081 = 4.26%. The chance on the next card is k / 47, here 2.13%.
The rule of 4 figure is 4 x k, here 4%, and its error is that figure minus the exact chance in percentage points, here -0.26 points.
Read the figures as a card count. All 47 unseen cards stay in, including any your opponent holds, so the figures describe your outs, not a matchup. Bets are called in full against a $100 pot with no extra money assumed.
Computed from the closed form by the GeorgeLiPoker math library.
Updated
Questions
How good is a draw with 1 out on the flop?
With 1 out on the flop you hit by the river 4.26% of the time, about 4 of every 100 hands, and on the next card 2.13%. The odds against hitting by the river are 22.5 : 1, since 1,035 of 1,081 pairs of unseen cards hold no out.
Is the rule of 4 close enough for 1 out?
The rule of 4 gives 4% for 1 out against an exact 4.26%, an error of -0.26 points. That is close enough to use at the table.
When should I call or fold with 1 out on the flop?
At 4.26% by the river, you can call none of the 12 bet sizes. A half-pot bet needs 25.00%, a margin of -20.74 points, so against half the pot the verdict is Fold. The largest bet you can call is 4.65% of the pot, or $4.65 on a $100 pot.
How many points do I lose if outs are shared with a better hand?
Take it off the count first. With your one out shared, the chance by the river falls from 4.26% to 0.00%, a loss of 4.26 points, and the effective count is then 0 outs.
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