KEY TAKEAWAYS
In the hand A♠ A♦ against 9♣ 8♣ on a J♥ 10♦ 4♠ flop, the overpair wins 67.4% of the time and the open-ended straight draw wins 32.6%, with no ties. These are exact figures from counting all 990 possible turn and river combinations, not estimates. The overpair is a clear favorite, but a one-in-three chance for the draw is far from small.
This article follows the hand street by street: the exact flop equity and where it comes from, how it changes with each possible turn card, and how the draw’s equity compares with the price of a call at several bet sizes. The hand is hypothetical and the cards are chosen so that every number can be rebuilt by hand. As with every equity, the figures describe the long run over many deals and do not predict the next card.
The Matchup and the Setup
An overpair is a pocket pair higher than every card on the board. Here the aces beat the jack and ten on the flop. The opponent holds 9♣ 8♣, which together with the J♥ 10♦ on the board makes four connected ranks, 8-9-10-J. That is an open-ended straight draw: a 7 or a queen completes a straight at either end.
The draw has 8 outs, the four sevens and the four queens, none of which is already visible. The aces have a faint extra route of their own: a king and a queen together would give them A-K-Q-J-10, a broadway straight. That detail matters for the exact count.
Because both hands are known in this analysis, the unseen cards are different from a player’s usual view. Seven cards are known (two aces, two draw cards, three board cards), leaving 45 unseen cards and C(45,2) = 990 equally likely turn-and-river combinations. In a real hand each player sees only their own two cards and the board, so 47 cards are unseen from either seat. The 45-card count applies only because both hands are shown.
How the Exact Equity Is Calculated
The reliable method is enumeration: list every one of the 990 runouts, decide the winner of each, and count. The method is described in general terms in how equity calculators work, and it can be done here with a short argument rather than a computer.
Step 1: Count the runouts where the draw completes a straight
The draw completes if at least one of the 8 outs appears. The runouts with no 7 and no queen are drawn from the 37 other cards, so there are C(37,2) = 666 of them. The runouts with at least one out number 990 − 666 = 324, which is 32.7%. The quick estimate from the rule of 2 and 4 is 8 × 4 = 32%, close to this figure.
Step 2: Remove the runouts where the aces make broadway
If a queen and a king both arrive, the aces make A-K-Q-J-10, which beats the draw’s queen-high straight. That is 4 queens × 4 kings = 16 runouts that the straight count wrongly credited to the draw.
Step 3: Add the runouts where the draw wins without a straight
The draw also beats the aces with two pair or trips. Both a 9 and an 8 on the board give 9s and 8s, so 3 × 3 = 9 runouts. Two nines give trips, 3 runouts, and two eights give trips, 3 runouts. That is 9 + 3 + 3 = 15 runouts.
The draw’s total is 324 − 16 + 15 = 323 runouts, or 323 ÷ 990 = 32.6%. The overpair wins the other 667, or 67.4%.
Worked Example: Equity Street by Street
On the flop, the equities are 67.4% for the aces and 32.6% for the draw. Once a turn card is dealt, one card remains and 44 river cards are possible. The table groups the 45 possible turn cards into five types and gives the aces’ equity for each.
| Turn card | Number of cards | Aces win (of 44 rivers) | Aces’ equity | Draw’s equity |
|---|---|---|---|---|
| Any 7 | 4 | 0 | 0.0% | 100.0% |
| Any queen | 4 | 4 | 9.1% | 90.9% |
| Any 8 or 9 | 6 | 31 | 70.5% | 29.5% |
| Any king | 4 | 40 | 90.9% | 9.1% |
| Any other card | 27 | 36 | 81.8% | 18.2% |
The rows check out against the flop figure. The total number of winning runouts for the aces is 4 × 0 + 4 × 4 + 6 × 31 + 4 × 40 + 27 × 36 = 1,334 out of 45 × 44 = 1,980 ordered runouts, and 1,334 ÷ 1,980 = 67.4%, matching the flop equity exactly.
The rows also show why equity moves so much. A 7 turns the hand into a certain loss for the aces, because the draw already holds the straight and no river card can beat it. A queen gives the draw a straight too, and the aces survive only if a king arrives (4 of 44 rivers). A king helps the aces, because it sets up their own broadway if a queen follows, and only a 7 on the river (4 of 44) rescues the draw. On a blank turn the draw still has 8 outs among 44 cards, which is 8 ÷ 44 = 18.2%.
Comparing the Draw’s Equity With the Price
Equity becomes more useful when it is set beside the price of a call. This section only compares numbers; it does not say what anyone should do. Suppose the flop pot is 100 and the aces bet. The draw’s break-even equity is the call divided by the pot after the call, as in pot odds. The table uses the draw’s 32.6% flop equity.
| Bet into pot of 100 | Draw’s call | Equity needed | Draw’s equity minus needed |
|---|---|---|---|
| 25 | 25 | 16.7% | +16.0 points |
| 50 | 50 | 25.0% | +7.6 points |
| 75 | 75 | 30.0% | +2.6 points |
| 100 | 100 | 33.3% | −0.7 points |
Up to a three-quarter-pot bet, the draw’s equity exceeds the break-even figure. At a pot-sized bet it falls 0.7 points short, so the price is almost, but not quite, covered by equity alone.
The picture changes on a blank turn. With a pot of 200 and a bet of 100, the draw must call 100 into 300, so the call needs 100 ÷ (300 + 100) = 25%, while its equity is only 18.2%. The extra winnings needed to break even are 100 × (36 ÷ 8) − 300 = 150 chips, the figure explained in implied odds. Those 150 chips are a requirement, not a forecast, because later betting is uncertain.
Common Mistakes in This Kind of Analysis
The first mistake is treating “8 outs” as the full answer. Eight outs give 8 × 4 = 32% by the rule, which happens to be near the exact 32.6%, but only because three adjustments (324 runouts, minus 16, plus 15) nearly cancel out. In another hand the same shortcut can be several points off.
A second error is using 47 unseen cards when both hands are known. When both hands are shown, 45 cards are unseen and 990 runouts exist. Using 47 (the single-player view) gives a different, wrong answer.
Third, counting every queen as a clean out ignores that a queen on the turn lets the aces make broadway with a king. Some of the draw’s outs are partly tainted: after a queen, the 4 kings among the 44 river cards still give the aces the win.
Finally, an equity figure is easily misread as a prediction. A 67.4% favorite still loses almost one deal in three, and a single result says nothing about the equity itself. Treating a favorite as safe is as wrong as treating a draw as hopeless.
Where This Hand Fits in Real Play
Hand-versus-hand analysis is a teaching tool, because a real opponent’s cards are unknown. In practice a player compares a hand with a range of possible hands, and the equity of an overpair against a whole range is different from its equity against one hand. The figures here describe one specific matchup, and nothing in them should be copied onto a different board or a different pair of hands without recalculating.
Board texture also matters. The same aces against the same draw on a flop with a different suit pattern would give a different answer, since flush possibilities change the outs. Everything on this page describes averages over many repetitions. Poker involves real financial risk, short-term results depend heavily on luck, and the legal age and legality of real-money poker vary by state and venue. Knowing the math does not remove variance.
The Next Step in Hand Analysis
A different confrontation, with a different structure, is worked through in the set against a flush draw comparison, where the leading hand has a set instead of an overpair. For the tools used to check numbers like these, see how equity calculators work. Together, the two articles show how hand-versus-hand equity changes with the type of hand, and why exact enumeration is the reference against which every shortcut is checked.
