Preflop Odds Heads Up 4

[SS] “Okay, one last preflop matchup trivia question”, Stan the Stat promised. “You have almost no chance on this one, so I’ll give you the answer after you each take just one guess… What nontrivial preflop all-in matchup is the closest to a coin flip?”

[FF] “I always thought a pair of Queens against Ace-King was close”, volunteered Figaro the Fish.

[HH] “I’d go with something near what you mentioned earlier”, Harriet the Hazy suggested. “Like Jack-Ten offsuit against a pair of Twos.”

[RR] “Or the Queen-Jack suited against Twos”, Roderick the Rock recommended.

[LL] “Maybe something lower like Eight-Seven suited against the same”, Leroy the Lion offered.

[EE] “Yeah, it’s probably something like that”, Elias the Eagle amended. “I’ll guess the same but against Threes.”

[TT] “Since I get to guess for free / How ’bout the suited Four-Three / Against the ultimate poo / The offsuit Seven and Two?” Tyrone the Telephone attempted.

[SS] “All excellent guesses… well, maybe not Figaro’s, which is 54% suited. Tyrone was very close to the sixth best matchup: 62o vs. 54o where the Two matches a suit (the high card or kicker value of the Six almost exactly balances the straight potential of the 54). Fifth is 73s vs. 22 in different suits, and fourth is 97o vs. 22 with four suits. Third is what you all danced around, QJo vs. a pair of double suit-dominated Threes. Roderick just missed that one. Second is T9o vs. 55 with one matching suit. All of these are 50.01% for the better hand, as is number one, which wins by a smidge: ATs vs. 33 with one matching suit, which is so close to a coin flip that if you played it out 7,075 times, you’d only expect to win one more time than your opponent (50.0071%)! In all the cases with pairs, the pair wins with more sets and boats but loses to more straights and, perhaps surprisingly, pairs (and even two pairs sometimes because of the dreaded three-pair hands).”

[RR] “Very cool, Stan”, Roderick the Rock acknowledged. “But now that you’ve given us all of these mostly non-nutritious snacks, what about the meat and potatoes of all-in heads-up matchups?”

[SS] “I was getting to that”, Stan the Stat claimed. “When I was first learning how to play Hold ‘Em, I set out to memorize all of the common odds. I thought preflop all-in percentages would be useful, but there were just way too many to remember1. Fortunately, grouping the matchups into just eight general categories with their approximate odds is quite sufficient.”

[SS] “If neither hand is paired, there are three groups of matchups:”

Opposing Hand Equity Example2
Unpaired Dominated 70%3 KQo vs. K8o (75%)
QTo vs. JTo (73%)
Two Undercards
or Alternating Ranks4
65%5 AJo vs. 63o (65%)
QTo vs. J9o (64%)
Tweeners 57%6 A9o vs. QJo (56%)

[SS] “Otherwise with a pair, there are five matchup groups:”

Opposing Hand Equity Example
Dominated With Undercard 90% KK vs. KQo (91%)
Unpaired Undercards 85%7 QQ vs. 94o (87%)
Lower Pocket Pair 81% AA vs. KK (82%)8
One Overcard,
Possibly Dominated
70% QQ vs. K8o (72%)
KK vs. AKo (70%)
Two Overcards 55%* 44 vs. A7o (55%)
44 vs. QJo (51%)
44 vs. QJs (49%)9

[SS] “* Within a given category (when relevant), being suited is worth a few percent for the flushes (just being able to make a winning flush is worth half a percent), and being connected is worth a few percent for the straight possibilities. In cases where the flush or straight is one of the few ways to win, the difference for the weaker hand can be up to five percent (e.g., AA vs. AKs is 5% better than AA vs. AKo, and AA vs. T9o is 5% better than AA vs. T5o).”

[RR] “Why isn’t it always five percent?”

[SS] “All the hands where the weaker hand hits a straight or flush but would have had a winning pair, two pairs, or three of a kind anyway don’t increase the percentages. The straight or flush is superfluous in those cases. Similarly, suited connectors don’t get the full gain for both the straight and flush possibilities, more like just seven percent in the best cases.”

[TT] “If you’re all-in, while nothing’s been fated / Know the odds, lest your hopes get inflated / It’s better not to be dominated / Or for ‘Next Bust’ you’ll be nominated”, Tyrone concluded.

Footnotes:

  1. The total number of possibilities is 812,175 (52-choose-2 * 50-choose-2 / 2), but ignoring suits, there are only 14,196 (13^2-choose-2) to memorize.
  2. These charts are based on the tables in John Vorhaus’s Killer Poker by the Numbers, pages 268-275, but have been modified with help from Mathematrucker. All odds are approximate, within a couple of percent except as noted in some of the following footnotes.
  3. The full range is fairly wide, going from about 65% to 78%, with higher cards tending toward the upper part of the range.
  4. Having alternating ranks (e.g, QTo vs. J9o) makes surprisingly little difference (the weaker hand mostly needs to pair up in either case, and having two undercards can give more straight possibilities).
  5. The full range is fairly wide, going from about 58% to 71%.
  6. The full range goes from about 51% to 60%.
  7. The full range is fairly wide, going from 76.93% (KK vs. 54s) to 90.10% (AA vs. K2o with matching suits).
  8. The full range only goes from 79.75% (33 vs. 22 of different suits) to 82.69% (TT vs. 99 with matching suits). Vorhaus listed this as 80%, probably from rounding, but it’s definitely closer to 81%.
  9. The classic race, QQ vs. AKo (57%) or vs. AKs (54%), would also go here as the AK is effectively unconnected with only two Queens left in the deck to make a royal straight.

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