KEY TAKEAWAYS
CONTENTS
In this classic confrontation, the set is a 75.4% favorite and the flush draw wins 24.6%. The hand is a set of eights (8♣ 8♦) against a nut flush draw (A♥ K♥) on a flop of 8♥ 7♥ 2♠. Counting every possible turn and river, the set wins 746 of the 990 runouts and the flush draw wins 244, with no ties.
Many articles quote a flush draw as “about 35%” against any made hand. That number is the chance of hitting a flush with 9 clean outs, and it does not hold here: the set can fill up, so some hearts do not win. The gap between 35% and 24.6% is the main lesson of this matchup.
This guide follows the hand through each street. It shows how the equity is counted, how it moves when the turn comes, which cards matter most, and which calculation mistakes cause the 35% error. Every figure comes from enumerating the cards still to come, and the arithmetic is shown so it can be checked.
The Matchup: A Set Against a Flush Draw
A set is three of a kind made with a pocket pair and one matching board card. A flush draw is four cards of one suit that need a fifth. Here, the set player holds 8♣ 8♦ and the board gives the third eight (8♥). The draw player holds A♥ K♥, and with 8♥ 7♥ on the board has four hearts.
The set is currently ahead because three eights beat ace-high. The flush draw has not made a hand: it holds only ace-high and needs a heart. Each player’s equity is the share of all remaining cards that leaves them with the best hand at showdown. That is a flop equity if both turn and river are still to come, and a smaller, one-card figure once the turn is dealt.
The unseen cards matter. Between the four hole cards and three board cards, 7 cards are visible, so 45 cards remain. The number of possible turn-and-river pairs is C(45,2) = 990, which is the denominator for every flop equity figure below.
Why a Flush Draw Is Not 35% Against a Set
Start with the draw’s outs. There are 13 hearts in the deck, and A♥, K♥, 8♥ and 7♥ are visible, so 13 − 4 = 9 hearts remain. The chance that at least one arrives across two cards is 1 − (36 ÷ 45) × (35 ÷ 44) = 1 − 1,260 ÷ 1,980 = 36.4%, or 360 of the 990 runouts. This is the familiar figure, and it is where the quick “about 35%” comes from.
The flaw is that the draw only wins if the flush beats the set, and not every heart is a clean out. The 2♥ pairs the board and gives the set player 8-8-8-2-2, a full house that beats any flush. On a blank turn such as 3♠, the 3♥ on the river does the same thing. Among the 360 runouts containing a heart, the set still wins 116 of them, and the flush wins the other 244.
This is what the method in counting outs warns about: a clean out gives you the best hand, and a tainted out does not. The rule of 2 and 4 would give 9 × 4 = 36% for the draw. As an approximation it ignores tainted outs, and in this matchup it overstates the draw by about 11 percentage points.
Worked Example: Equity on the Flop and the Turn
The flop equity comes from counting all 990 runouts. The set wins 746, so its equity is 746 ÷ 990 = 75.4%. The flush draw wins 244, so its equity is 244 ÷ 990 = 24.6%. Adding them gives 100% because no runout produces a tie.
The turn then splits the 45 unseen cards into four groups. Seven cards (2♦, 2♣, 7♠, 7♦, 7♣, 8♠ and 2♥) leave the set with a full house or four of a kind, so it cannot lose. Twenty-four non-heart cards are blanks: the set is then 37 ÷ 44 = 84.1% because 7 of the 44 river cards are clean hearts. Six aces and kings give the draw a pair, and the set is 36 ÷ 44 = 81.8%. Eight clean hearts make the flush, and the set falls to 10 ÷ 44 = 22.7%.
The weighted average checks out: (7 × 100% + 24 × 84.1% + 6 × 81.8% + 8 × 22.7%) ÷ 45 = 75.4%, which matches the flop figure.
How the Numbers Shift From Street to Street
The table shows the set’s equity after each possible turn card and how many of the 45 cards produce that result.
| Turn card | Cards | Set equity | Flush draw equity |
|---|---|---|---|
| Flop (before the turn) | 45 | 75.4% | 24.6% |
| Blank (3, 4, 5, 6, 9, 10, J, Q of a non-heart suit) | 24 | 84.1% | 15.9% |
| Ace or king (not a heart) | 6 | 81.8% | 18.2% |
| Clean heart (3♥, 4♥, 5♥, 6♥, 9♥, 10♥, J♥, Q♥) | 8 | 22.7% | 77.3% |
| Pairs the board for the set, or 2♥ | 7 | 100.0% | 0.0% |
The turn swings equity sharply, not gradually. A flush draw going from 24.6% to 77.3% on a heart is a 52.7-point jump, while a blank leaves it at 15.9%. That uneven movement is why equity on the flop is an average of very different futures.
The set also improves on its own. It reaches a full house or better in 341 of 990 runouts, or 34.4%, which is close to the 36.4% of runouts that contain a heart. Seeing both sides of the matchup is what keeps the analysis honest, because a draw is not the only hand that can improve.
Common Mistakes in Set vs. Flush Draw Math
The first mistake is using 35% for the flush draw. The figure treats all 9 hearts as clean and uses 47 unseen cards, as if the opponent’s two cards were hidden. In this matchup, the right number is 24.6%, not 35%.
The second is treating every flush card as a winning card. Hearts that pair the board hand the set a full house, so they must be removed from the count. The third is forgetting that the set can improve: 7 of the 45 turn cards leave the set unbeatable, and others help it by the river.
A fourth is generalizing one hand. Change the set’s rank or the board, and the number changes. A bottom set against a different draw would not give 75.4%. The final mistake is reading equity as a prediction. A 75.4% favorite still loses almost one time in four, and nothing in the math says this particular hand will follow the average.
Comparing Equity With the Price of a Call
Equity becomes more meaningful when set against the price of continuing, which is what pot odds measure. Suppose the pot is 100 and the set player bets 75 on the flop. The flush draw must call 75 to win a pot of 250 after the call, so the price is 75 ÷ 250 = 30.0%. The draw’s equity is 24.6%, so on direct odds the call is below break-even.
Now suppose the turn is a blank, and the pot is 250 after a called flop bet. A bet of 125 means the draw needs 125 ÷ (375 + 125) = 25.0%. The draw’s equity on the river is 7 ÷ 44 = 15.9%. The gap is larger than before because one card is left and the draw has only 7 clean hearts.
Those comparisons are arithmetic about one hypothetical hand. They do not include chips that might change hands on later streets, and they do not say what any player should do. Poker involves real financial risk, and every equity figure describes a long-run average over many repetitions of the same situation.
Where the Same Method Applies Next
The same method applies to other classic confrontations: an overpair against a straight draw, two overcards against a pair, or a flush draw against a smaller flush draw. Each one begins by listing the cards, counting the unseen ones, identifying clean and tainted outs, and checking the result against the full count of runouts. Rules of thumb are useful for estimates, but the enumeration here shows how far a shortcut can drift when the outs are not clean.
