Verdict
Fold
1 out, flop, pot-size bet
- Chance to hit by the rivertwo cards to come4.26%
- Equity the call needs$100 into a pot of $30033.33%
- Margin over the priceyour chance minus the price-29.08 points
The call in numbers
4.26% against 33.33%: the draw trails the price by 29.08 points. On this flop against a pot-size bet, that is an easy fold.
Holding a single out means one card in the deck helps you. Out of 47 unseen cards it arrives 2.13% of the time on one card.
Counting the chips on a $100 pot
- Starting pot$100
- Your opponent bets$100
- Chips you put in$100
- Pot with your call in$300
- You can win on a hit$200
- What a miss costs$100
- Your chance to hit4.26%
- Average value of the call-$87.23
- Break-even equity33.33%
- Your chance minus the price, unrounded-29.0780 points
- Fractional outs at the price8.53 outs
- Outs above the price-7.53 outs
Price it in dollars, with no later bets: win the whole pot on a hit, lose the call on a miss. That averages $8.51 in and $95.74 out, so calling loses $87.23 on average at -0.872 per dollar called.
In odds, you face 22.5 : 1 against you and the pot pays 2 : 1.
Counting the pot without your call makes the price look like 50.00%, when the real price is 33.33%. On that mistaken price the margin would read -45.7447 points.
The price with implied odds
The call becomes fair only if a hit brings in $2,050 more after the draw arrives. Measured against the $100 you pay, that is 20.50 times your call.
Outs are the other lever. At 9 outs you reach 34.97% by the river, enough for a pot-size bet, and you are 8 outs away.
| Future winnings | Price with them | Your chance minus price | Value of the call | Call? |
|---|---|---|---|---|
| $0 | 33.33% | -29.08 points | -$87.23 | Fold |
| $50 | 28.57% | -24.32 points | -$85.11 | Fold |
| $100 | 25.00% | -20.74 points | -$82.98 | Fold |
| $250 | 18.18% | -13.93 points | -$76.60 | Fold |
| $500 | 12.50% | -8.24 points | -$65.96 | Fold |
The next card against the whole runout
Both views point the same way. By the river you hit 4.26%, on the next card 2.13%, and the price is 33.33%, so the fold holds whichever number applies.
| Cards counted | Chance | Margin over price | Value of the call | Verdict |
|---|---|---|---|---|
| By the river, two cards | 4.26% | -29.0780 points | -$87.23 | Fold |
| Next card only | 2.13% | -31.2057 points | -$93.62 | Fold |
Draws around this one
One out can change the answer, so the table lists the counts just above 1. Each out moves the chance roughly 4.16 points against the 33.33% price.
Every bet size, one draw
Your 4.26% chance is fixed, so the only thing that moves is the price. The 12 rows below price the call at every bet size on the flop.
Your draw can face a bet of up to 0.05 times the pot ($4.65 on a $100 pot) and still break even. That is under the quarter-pot bet, so none of the 12 sizes is a call.
| Bet size | Equity needed | Margin | Verdict |
|---|---|---|---|
| quarter-pot | 16.67% | -12.41 points | Fold |
| third-pot | 20.00% | -15.74 points | Fold |
| two-fifths-pot | 22.22% | -17.97 points | Fold |
| half-pot | 25.00% | -20.74 points | Fold |
| two-thirds-pot | 28.57% | -24.32 points | Fold |
| three-quarter-pot | 30.00% | -25.74 points | Fold |
| pot-size, this page | 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 |
How this is computed
Everything is computed from the closed form. Pairs without an out: 1,035 of 1,081. The rest, 4.26%, hold at least one out.
The model is deliberately simple: call everything, win everything on a hit, face no more bets, deal from 47 equally likely unseen cards, and ignore an opponent's holding. With the price at $100 / $300 = 33.33%, later streets and shared outs are what change the real answer.
Computed from the closed form by the GeorgeLiPoker math library.
Updated
Questions
What is the right play with 1 out when a pot-size bet comes on the flop?
Compare 4.26%, your chance to hit by the river, with 33.33%, the equity a pot-size bet needs. The gap is -29.08 points, so this is a fold. In dollars on a $100 pot, the call averages -$87.23 when no more betting follows.
At what count of outs does a pot-size bet on the flop become a call?
A call starts to work at 9 outs: 34.97% to hit versus 33.33% needed. One out fewer drops the chance to 31.45%. You hold 1, which is 8 fewer than the minimum.
Should you use the chance on the next card or by the river?
It depends on what follows, but here it does not matter. If the hand ends here, count both cards: 4.26%. If another bet is coming on the turn, count one card: 2.13%. Against 33.33%, both say fold.
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