Three Hearts, Half the Draw

October 5, 2026 by LLSSAT

You look down at A♥ K♥ 7♥ on third street. Five players are still in the hand. You have two big cards and the makings of an ace-high flush. It is an attractive hand for a crowded stud pot.

Now remember the three players who just folded. Did they throw away clubs, or did you see the 9♥, T♥ and J♥ disappear?

Your own cards have not changed. Your draw has. In the controlled example below, those three exposed hearts almost cut the chance of completing a flush in half. The hand also collects a smaller share of the pot at showdown.

Our Counting Cards discussion explains why remembering folded cards matters. Here we use the new Stud High simulator to measure one example. This is high-only seven-card stud, with the best five of seven deciding the winner.

The same hand, four different memories

Eight players were dealt in. Three folded on third street, leaving you and four opponents. Your opponents' visible door cards are Q♠, 8♦, 3♠ and 2♦. None is a heart.

We keep those cards and your A♥ K♥ 7♥ fixed. The folded door cards always have the ranks nine, ten and jack. Only their suits change:

• No folded hearts: 9♣ T♣ J♣.
• One folded heart: 9♥ T♣ J♣.
• Two folded hearts: 9♥ T♥ J♣.
• Three folded hearts: 9♥ T♥ J♥.

The folded players' down cards stay unknown. We do not peek at anyone's concealed cards, add future upcards to the third-street decision, or give folded players more cards.

For each version, we sampled 200,000 legal showdowns against uniformly distributed unknown opponent cards. Everyone still active received their remaining cards and reached showdown. These are random-card opponents, not measured ranges for players who chose to continue. There was no later betting, folding or rake.

What the missing hearts cost

Swipe the table sideways to see the complete comparison.

The flush column is an exact card-counting calculation, including the rare straight flush. You already have three hearts and need at least two more among your four remaining cards. The simulator's sampled flush frequencies agreed with that separate calculation.

Remembering three folded hearts changes the completion chance from 23.56% to 12.29%. The hand's average share of the final pot falls by 8.86 percentage points, from 37.01% to 28.15%, in this five-player model.

Average pot share counts a whole-pot win as 100%, a two-way tie as 50%, and a loss as zero. It is often called showdown equity. It includes wins with pairs, two pair and other hands as well as flushes. It is not the chance of making a flush, and it does not measure betting profit.

Unseen is different from sitting in the deck

In each version, you know ten cards: your three, the four active door cards and the three folded door cards. That leaves 42 unseen card identities. With no other exposed hearts, ten of those identities are hearts; with three folded hearts, seven are.

Some unseen hearts may already be in opponents' down cards or in the muck. You do not know which ones. Calling them unseen is more accurate than promising that all ten are physically waiting to be dealt.

Our uniform calculation averages over those unknown locations. It also allows for unknown burns and the folded players' concealed cards. You should remove the cards you actually saw, rather than inventing identities for cards you could not see.

The count can change again on fourth street. A newly exposed heart in another player's hand is new information then. It was not information available when you judged the hand on third street.

A draw and a winning hand are different questions

We made a second check with only you and the Q♠ opponent active. The 8♦, 3♠ and 2♦ players had already folded too, so the same ten known cards remained excluded. This keeps the information about your flush draw the same while changing the number of hands eligible to win.

The flush chances are unchanged, but the pot shares are much larger. Fewer hands are competing with your pairs and other improvements, or trying to beat your eventual flush. A useful draw count is only one part of judging your hand.

We also changed the three discarded clubs in the original no-heart case to diamonds. Its five-player pot share was 37.17%, with an interval of 36.87-37.46%, compared with 37.01% in the club case. The same exact 23.56% flush chance applied. This control gave a similar result without relying on one particular non-heart suit.

What to take back to your game

Count the exposed cards before admiring your own. An ace-high three-flush with its suit mostly unseen deserves a different assessment from the same cards after several matching upcards have disappeared.

Start by remembering what helps your hand. For this draw, note the hearts as players fold, then the aces and kings that affect your other routes to improvement. Keep reading the active boards; folded cards remain dead even after the hand showing them has gone.

Keep the price and the players in view. A large multiway pot can make a draw worth pursuing. Later bets and raises can also make it expensive, and real opponents select which hands they play. These percentages do not establish whether calling or raising is profitable in your game.

The practical lesson is to carry your memory into the next decision. You still hold three hearts after the other hearts are folded. You have fewer ways to turn those three hearts into a winning five-card hand.

Study details and downloads

Poker Simulator 0.16.0; ruleset stud-high-uniform-showdown-v1; implementation stud-high-research-v1. Third-street observer information only. Standard high, best five of seven, one common pot with equal fractional tie shares. The normal exposed/hidden dealing sequence is described in the PokerStars stud rules.

Each main case used two independent seeds and 100,000 trials per seed: 2026100511-12, 2026100521-22, 2026100531-32 and 2026100541-42 for zero through three folded hearts. Each control used a separate 100,000-trial seed. In total, 1,100,000 sampled showdowns were checked, together with 2,964 exactly enumerated completions in two separate fully known sixth-street matchup checks. Those known-matchup checks are validation fixtures, not additional knowledge used in the third-street tables.

The exact flush calculation counts four-card runouts containing at least two hearts among the 42 unseen identities. Equity intervals use pooled sample variance and a normal 95% approximation. They describe Monte Carlo noise under the fixed uniform assumptions, not uncertainty about real opponents. Different discarded suits change the available cards for both you and your opponents; the deals across versions were not paired. No betting model, future folds, calibrated ranges, rake, all-ins or side pots are included.

Download the results table (CSV), study summary and checks (JSON), or complete reproduction package (ZIP). The package includes exact inputs, full reports, retained audit traces, seeds, checksums, commands and a Python script for rerunning the study with Poker Simulator 0.16.0.