Chance to hit by the river
8.42%
2 outs, two cards to come
- Chance on the next card4.26%
- Odds against by the river10.88 : 1
- Rule of 4 estimate-0.42 points from the exact chance8%
The count behind 2 outs on the flop
8.42% is a modest chance. Over many hands the draw lands about 8 times in 100, and that is rarely enough for a large bet.
- Unseen cards47
- Your outs2
- Cards that miss45
- Pairs of cards with no out990
- Pairs with at least one out91
- Chance by the river8.42%
- Chance on the next card4.26%
- Odds against hitting10.88 : 1
- Estimate from the rule of 48%
- Error of the rule of 4-0.42 points
Chance by the river = 1 - C(u - k, 2) / C(u, 2), with u = 47 unseen cardsFor 2 outs: 1 - C(45, 2) / C(47, 2) = 1 - 990 / 1,081 = 8.42%Next card = 2 / 47 = 4.26%
Count the ways to miss twice: 45 cards miss, and 990 pairs can be drawn from them. Take that from 1,081 pairs in all and 91 are left, so 8.42% of the run-outs help you.
With 2 outs the rule of 4 stays near the truth: 8% estimated, 8.42% exact, 0.42 points apart.
Leave the adjusted rule for bigger flop draws. With 2 outs on the flop, the plain rule of 4 figure is 8%.
Which holding has 2 outs?
One standard source of this count is a pocket pair that wants a set. Count the cards carefully at the table, because outs that also help a rival do not count in full.
Not every out is clean. Here is the chance once shared outs are removed: from 8.42% down to 0.00% when 2 are shared.
| Shared outs | Effective outs | Chance by the river | Chance given up |
|---|---|---|---|
| 0 | 2 | 8.42% | 0.00 points |
| 1 | 1 | 4.26% | 4.16 points |
| 2 | 0 | 0.00% | 8.42 points |
The bets 2 outs 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 biggest bet you can call is 10.12% of the pot, which is $10.12 on a $100 pot.
| Bet size | Equity a call needs | Margin over it | Verdict |
|---|---|---|---|
| 1/4 pot | 16.67% | -8.25 points | Fold |
| 1/3 pot | 20.00% | -11.58 points | Fold |
| 2/5 pot | 22.22% | -13.80 points | Fold |
| 1/2 pot | 25.00% | -16.58 points | Fold |
| 2/3 pot | 28.57% | -20.15 points | Fold |
| 3/4 pot | 30.00% | -21.58 points | Fold |
| full pot | 33.33% | -24.92 points | Fold |
| 1.25x pot | 35.71% | -27.30 points | Fold |
| 1.5x pot | 37.50% | -29.08 points | Fold |
| 2x pot | 40.00% | -31.58 points | Fold |
| 3x pot | 42.86% | -34.44 points | Fold |
| 4x pot | 44.44% | -36.03 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 4.07 points on this street.
- 2 outs on the turn: 4.35% on the river.
- 1 out on the flop: 4.26% by the river.
- 3 outs on the flop: 12.49% by the river.
When a bet buys only one card
When another bet is coming on the turn, a call on the flop buys you one card, not two. Your next-card chance is 4.26%, against 25.00% needed to call a half-pot bet, a margin of -20.74 points, so the next-card verdict is 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 2 outs that is 1 - 990 / 1,081 = 8.42%. The chance on the next card is k / 47, here 4.26%.
The rule of 4 figure is 4 x k, here 8%, and its error is that figure minus the exact chance in percentage points, here -0.42 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
What percent chance does a count of 2 give you on the flop?
With 2 outs on the flop you hit by the river 8.42% of the time, about 8 of every 100 hands, and on the next card 4.26%. The odds against hitting by the river are 10.88 : 1, since 990 of 1,081 pairs of unseen cards hold no out.
Can I trust the shortcut for 2 outs on the flop?
The rule of 4 gives 8% for 2 outs against an exact 8.42%, an error of -0.42 points. That is close enough to use at the table.
When should I call or fold with 2 outs on the flop?
At 8.42% by the river, you can call none of the 12 bet sizes. A half-pot bet needs 25.00%, a margin of -16.58 points, so against half the pot the verdict is Fold. The largest bet you can call is 10.12% of the pot, or $10.12 on a $100 pot.
Does every one of the 2 outs count in full on the flop?
Take them off the count first. With one shared out the chance by the river falls from 8.42% to 4.26%, and with 2 shared it falls to 0.00%, a loss of 8.42 points. The effective count is then 0 outs.
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