How often does a set improve to a full house?
Outs
A set on the flop improves to a full house or better by the river 33.4% of the time, not the 27.8% a plain count of seven outs suggests.
- 14.9%improves on the turn
- 21.7%improves on the river after a blank turn
- 2.0 to 1odds against improving by the river
The True Probability of Improvement
A set on the flop improves to a full house or better by the river in roughly one-third of cases. The table below shows the specific chances for each street. A simple count of seven outs yields a lower figure than the actual probability. This discrepancy arises because the number of available outs changes depending on whether the turn card has already been dealt. The chance to hit on the river is higher than on the turn because fewer unseen cards remain in the deck. This conditional increase compensates for the initial miss, raising the total probability above the naive estimate.
| A set on the flop | Chance to improve | Odds against |
|---|---|---|
| Turn, one card | 14.9% | 5.7 to 1 |
| River after a blank turn | 21.7% | 3.6 to 1 |
| Turn or river | 33.4% | 2.0 to 1 |
Why Seven Outs Underestimate the Odds
The standard rule of multiplying outs by four assumes a fixed number of opportunities. In reality, the probability of hitting on the river depends on the outcome of the turn. If the turn misses, the remaining deck is smaller, concentrating the remaining outs. This increases the likelihood of success on the final card compared to a static calculation. The combined effect of a moderate chance on the turn and a higher conditional chance on the river results in a total probability that exceeds the simple sum. The table above illustrates how the river chance surpasses the turn chance, driving the overall figure higher than expected.
Comparing Street-by-Street Chances
The chance to improve on the river after a blank turn is larger than the chance to improve on the turn itself. This occurs because the pool of unknown cards shrinks, making each remaining out more potent. The total probability of improving by the river is larger than the chance of improving on the river alone. It is also larger than the naive estimate derived from a fixed seven-out count. These relationships show that time adds value to the hand, provided the board does not pair early. The odds against improving by the river reflect this cumulative advantage over single-street calculations. If gambling stops being fun, free help is available via support resources.
Questions
Why is the river chance higher than the turn chance?
After a blank turn, fewer cards remain in the deck. This concentrates the remaining outs, increasing the probability that the next card completes the hand compared to the initial turn opportunity.
Does the number of outs change during the hand?
Yes. While the initial count is fixed, the effective probability shifts as cards are dealt. A miss on the turn improves the conditional odds for the river, raising the total success rate.
Is the simple rule accurate for this scenario?
No. The basic formula underestimates the total probability because it does not account for the increased density of outs on the river after a miss. The actual chance is higher.
Every figure on this page is computed by code from a standard 52-card deck, counting every possible card and checking the count a second way. See the methodology.