Wraps and the Real Numbers

Three shapes of wrap

A wrap is a straight draw built from three or four of your cards around the board's connectors. How many outs it has depends entirely on where your cards sit relative to the board, and the difference between shapes is enormous. All three below are on a connected flop:

• 7♣6♥5♦4♠ on 9♥8♦2♣ — 13 outs, 3 of them nut outs. All four of your cards sit below the board's connectors.
• J♠T♥7♦2♣ on 9♥8♦5♣ — 17 outs, 11 nut outs. Two cards above the connectors, one sitting in the gap between them.
• J♠T♥7♦6♣ on 9♥8♦2♣ — 20 outs, 14 nut outs. Two above and two below: the board's connectors are wrapped on both sides.

Cards on both sides of the board's connectors give the most outs, and the cards above the board are the ones that make those outs nut outs.

On an unpaired flop with three suits, when you have not already made a straight. It is a statement about shape, not about how pretty the hand looks.

J♠T♥6♣5♣ on 9♥8♦2♣ is also four connected cards, on the same flop, and it has 8 outs. The six and the five are below the eight and simply do not participate. Four connected cards is not the criterion; where they sit is.

Hitting is not winning

Take the biggest wrap in the game, J♠T♥7♦6♣ on 9♥8♦2♣: 20 outs, 14 of them nut outs. Two numbers describe it, and they are not the same number:

• Hit probability 69.7% — the chance at least one out arrives, over all 990 run-outs.
• Equity 50.8% — the share of the pot you win against a specific opponent, here top set, averaged over all 2,625 hands that contain it.

The gap is 18.9 points, and it is not a rounding artefact. Some of the cards that improve you improve him more; the board can pair and give him a full house on a card that made your straight. In hold'em outs and win rate are close enough to convert between, which is why the habit exists. Here the opponent is drawing too.

A third number finishes the picture: of the times this hand does finish with a straight or better, only 35.5% of the time is it still the absolute best hand at showdown.

State a draw as hit probability and equity, separately, and never convert one into the other.

Every street. The two quantities answer different questions: hit probability is about the deck, equity is about a specific opponent or range.

More outs does not mean safer. J♠T♥6♣5♣ has only 8 outs and hits just 37.1% of the time — but when it hits, it is still the nuts 46.6% of the time, better than the twenty-out wrap's 35.5%. All 8 of its outs are nut outs; many of the big wrap's are only best for one street.

The average is not the hand you are holding

"About a coin flip against a set" is the sentence everyone repeats about the twenty-out wrap. The average really is 50.8%. But we enumerated all 2,625 versions of top set, and the answer ranges from 34.9% to 59.3% — a band 24.4 points wide, decided entirely by the two cards you cannot see next to his set.

The same is true one shape down. 17 outs against top set averages 45.7%, ranging 29.6% to 53.4%. Thirteen outs averages 38.8%, and the eight-out nut draw 28.0%.

"Roughly even" is a fact about the average. At the table you are facing one specific hand, and the honest statement is a range.

Whenever a number is quoted against a hand class ("a set", "two pair") rather than four named cards. Against four named cards the enumeration is exact and there is no band.

"Even against a set, so I am never in bad shape" fails on the low end of its own band: 34.9% is not a coin flip, and it is the same hand, the same flop and the same set. Nothing about your read was wrong — the number simply was never a single number.

Keep these three anchors and you can rebuild most of this camp from memory: 13 outs hits 49.9%, 17 outs hits 61.8%, 20 outs hits 69.7%.

What to take away

• Three anchors: 13 outs 49.9%, 17 outs 61.8%, 20 outs 69.7%.
• Hit probability and equity are two quantities. The best wrap in the game hits 69.7% and is worth 50.8% against top set — 18.9 points apart.
• Hitting and staying best are two more. The big wrap is still the nuts only 35.5% of the times it hits; the small all-nut draw manages 46.6%.
• Quote a band, not a point. Any number against a hand class hides one.

All of the above are rules of this course, not an equilibrium solution. What they are not yet is a decision: knowing you have 69.7% to hit does not tell you whether to bet. The next two lessons build the decision — first the price you are being offered, then the threshold.