Verdict
Fold
3 outs, flop, three-quarter-pot bet
- Chance to hit by the rivertwo cards to come12.49%
- Equity the call needs$75 into a pot of $25030.00%
- Margin over the priceyour chance minus the price-17.51 points
Comparing the draw with the bet
The price is 30.00% and your chance is 12.49%. You sit 17.51 points below the line, which is a wide gap, so you fold.
For one overcard to top pair, 3 outs is the number to use, and one card hits 6.38% of the time out of 47 unseen.
The $100 pot example
- The pot before betting$100
- The bet$75
- Cost to call$75
- Total pot after calling$250
- Net gain when you hit$175
- Loss on a miss$75
- How often you hit12.49%
- Expected value of calling-$43.78
- Equity needed to break even30.00%
- Margin, to four decimals-17.5116 points
- Fractional outs at the price7.59 outs
- Outs above the price-4.59 outs
Stay with simple rules: the whole pot on a hit, the call lost on a miss, no more money in. Under them calling loses $43.78 on average. The hit side averages $21.85, the miss side $65.63, and each dollar called returns -0.584.
In odds, you face 7.01 : 1 against you and the pot pays 2.33 : 1.
The correct pot to divide by includes your call. Without it the price reads 42.86% rather than 30.00%, and the margin reads -30.3687 points. With the starting pot alone it reads 75.00% and -62.5116 points.
When later betting could rescue the call
The call becomes fair only if a hit brings in $350.56 more after the draw arrives. Measured against the $75 you pay, that is 4.67 times your call.
The call starts to work at 8 outs, where the chance is 31.45%. Your 3 outs fall 5 outs below that.
| Future winnings | Price with them | Your chance minus price | Value of the call | Call? |
|---|---|---|---|---|
| $0 | 30.00% | -17.51 points | -$43.78 | Fold |
| $50 | 25.00% | -12.51 points | -$37.53 | Fold |
| $100 | 21.43% | -8.94 points | -$31.29 | Fold |
| $250 | 15.00% | -2.51 points | -$12.56 | Fold |
| $500 | 10.00% | +2.49 points | +$18.66 | Call |
One card or two
No conflict here: 12.49% for two cards and 6.38% for one both give a fold against the 30.00% price, so you need not decide which view applies.
| Cards counted | Chance | Margin over price | Value of the call | Verdict |
|---|---|---|---|---|
| By the river, two cards | 12.49% | -17.5116 points | -$43.78 | Fold |
| Next card only | 6.38% | -23.6170 points | -$59.04 | Fold |
How the answer moves with outs
Move up or down a few outs and watch the verdict. Every extra out is worth about 3.98 points on the flop, and the price stays at 30.00%.
The same draw against other bet sizes
How much does bet size matter? The price climbs from 16.67% to 44.44% across the list, while 3 outs stay at 12.49%.
3 outs stop paying at 0.17 times the pot ($16.65 on a $100 pot), below the smallest size listed. The call fails at all 12 bet sizes.
| Bet size | Equity needed | Margin | Verdict |
|---|---|---|---|
| quarter-pot | 16.67% | -4.18 points | Fold |
| third-pot | 20.00% | -7.51 points | Fold |
| two-fifths-pot | 22.22% | -9.73 points | Fold |
| half-pot | 25.00% | -12.51 points | Fold |
| two-thirds-pot | 28.57% | -16.08 points | Fold |
| three-quarter-pot, this page | 30.00% | -17.51 points | Fold |
| pot-size | 33.33% | -20.84 points | Fold |
| 1.25x-pot | 35.71% | -23.23 points | Fold |
| 1.5x-pot | 37.50% | -25.01 points | Fold |
| 2x-pot | 40.00% | -27.51 points | Fold |
| 3x-pot | 42.86% | -30.37 points | Fold |
| 4x-pot | 44.44% | -31.96 points | Fold |
How this is computed
Counting is exact: the pairs of unseen cards with no out number 946 out of 1,081, so hitting by the river has chance 12.49%.
Four assumptions sit behind the verdict: you call the full bet, a hit wins the entire pot, no money goes in after the call, and every one of the 47 unseen cards is equally likely. The price is $75 / $250 = 30.00%. No card is taken out for what an opponent might hold.
Computed from the closed form by the GeorgeLiPoker math library.
Updated
Questions
What does the math say about calling a bet of three quarters of the pot with 3 outs on the flop?
Reduce the question to a comparison of two numbers: 12.49% to hit and 30.00% to break even. The first minus the second is -17.51 points, which makes this a fold, and the call averages -$43.78 on a $100 pot if no more betting follows.
What count of outs clears the price of a three-quarter-pot bet on the flop?
The threshold is 8 outs. It gives 31.45%, clearing 30.00%, whereas 7 gives only 27.84%. Holding 3 puts you 5 below it, with a chance of 12.49%.
Should you use the chance on the next card or by the river?
Both figures are right for their own situation: 12.49% with no more betting, 6.38% when the turn brings a bet. Facing 30.00%, each one gives a fold.
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