Freerolls and Redraws

The same straight, a different fate

Flop T♥9♥8♣. You hold A♥K♥Q♦J♦. He holds Q♠J♣4♦5♠. Right now the two of you have the same hand: queen-jack playing with ten-nine-eight, the best possible hand on this board, and neither of you is ahead of the other by a single chip.

Now enumerate every way the turn and river can come — all 820 of them:

• You win 47.0% of them.
• You split 53.0%.
• He wins 0.0%. Not one board out of the 820.

Your share of the pot is 73.5%, his is what is left, and he cannot do anything about it. He is drawing dead to a tie. That is a freeroll, and it is the single most profitable pattern in this game because it costs him money on every street while he believes he has the nuts — which, at this moment, he genuinely does.

Rule of the game: a showdown hand uses exactly 2 hole cards and exactly 3 board cards. Two players can therefore hold the identical best hand while holding completely different futures — the two cards that are not playing right now are the ones that decide the next street.

The 820 boards are the complete flop run-out enumeration from js/core/plo-equity.js, the same engine behind every equity number in this camp; the outs come from js/core/plo-outs.js.

When you hold the current nuts, the next question is not "am I ahead". It is "if we are tied, which of us can still improve".

Any street where more cards are coming and your hand can be duplicated — straights above all, because a straight is made of board cards that everyone can see.

Having a redraw is not the same as being freerolled. Of your 9 flush outs only 4 leave you holding the best possible hand — the rest arrive on boards where somebody else can hold something better. The free roll here is real because he can never win, not because your redraw is clean.

How many of your nut flush outs are actually the nuts

Everybody knows the phrase nut flush draw. Almost nobody checks it. Here is what checking looks like — every out enumerated against every two-card holding an opponent could have.

Flop 9♥8♥2♠, you hold A♥K♥7♦2♣: the two highest hearts in the deck. You have 9 flush outs. Exactly 2 of them leave you with the best possible hand. 6 of the others arrive on a board where somebody can hold a straight flush, and the last one pairs the board.

Flop A♥8♥3♠, you hold Q♥J♥7♦2♣. Now the ace is on the board, so nobody can hold it. Again 9 flush outs, and this time only 1 of them is the nuts — and here 0 of the misses are about straight flushes; they are about the king, which somebody else can hold.

Never call a draw "the nut flush draw" from your hand alone. The nuts is a specific two-card combination on a specific board; whether you hold it is a question about the board, not about your ace.

Every two-tone flop. The shortcut is at its most dangerous on connected boards, where the thing beating you is a straight flush rather than a bigger flush.

We measured the shortcut instead of arguing about it. Across 900 sampled spots with at least one flush out, "I hold the ace of that suit, so this out is the nuts" was wrong 506 times out of 1208 — 41.9%, standard error 3 points, computed by treating each spot rather than each out as one independent sample. A rule that is wrong two times in five is not a rule.

Because you must use exactly 2 hole cards, the best possible flush is always one named pair of cards. That is why no shortcut based on a single card can work, and why our counter enumerates every opponent holding instead (js/core/plo-outs.js, docs/OUTS-SPEC.md).

Counting a combo draw without counting anything twice

Flop 9♥8♥2♣, you hold J♥T♥7♦6♣. Two questions people answer separately and then add up wrong: how many cards make you a straight, and how many make you a flush.

• Straight outs: 18.
• Flush outs: 9.
• Cards that do both — they make you a straight flush: 2.
• Total distinct cards that improve you to a straight or better: 25, not twenty-seven.

That is what a combo draw count has to look like: a card is counted once, and it counts as the better of the two hands it makes. Our counter reports the three parts separately and checks the arithmetic itself, because an error of one in each part cancels out in the total and nothing would look wrong.

Ask a draw two questions, in this order: how often does it arrive, and how often does arriving leave you with the best hand. The second one is the redraw question and it is the one people skip.

Flop, any texture. Both quantities are conditional on the run-out, so they say nothing about how the money goes in.

Arriving more often can mean arriving worse. That 25-out monster gets there 80.8% of the time and is still the best hand on only 28.6% of the times it gets there. Meanwhile 9♣9♠A♥K♥ on the same board — top set with 8 flush outs — arrives just 59.1% of the time and is best on 16.9% of those. Neither number is the one people quote, and the one people quote is the first one.

Which is why the useful word here is redraw: not "do I have outs" but "when my outs come, am I still the one being paid". The nut flush draw on 9♥8♥2♠ from the previous page — 9 outs, 2 of them nut outs — is a redraw. A second-nut flush draw is a way to lose a stack.

What to take away

• Two players can hold the same hand and different futures. Same straight, and he wins 0.0% of 820 run-outs while you take 73.5%.
• "Nut flush draw" is a claim about the board. Checked against every opponent holding, the ace shortcut was wrong 41.9% of the time.
• Count the draw properly. 9 flush outs, 2 of them nut outs, is a different hand from nine clean ones.
• A draw has a second number. The 25-out combo draw is still best on only 28.6% of the boards where it arrives.

Every number on these four pages is an enumeration or a seeded sample from our own engine, and every threshold is one of the rules of this course, not an equilibrium solution. Next: the other side of the same coin — what you can do with the cards you are not using.