Preflop Odds Heads Up

[SS] “If your only opponent is all-in with a pair of Aces, what’s the best hand you can hold?” Stan the Stat opened.

[FF] “The other two Aces”, Figaro the Fish observed.

[SS] “Besides that, of course.”

[RR] “I thought it was Ten-Nine suited, matching neither of the Ace’s suits”, Roderick the Rock opined.

[LL] “No, if that’s good then Nine-Eight suited has to be better. With Ten-Nine, a King-high straight is no good, since the Aces win with a royal straight”, Leroy the Lion objected.

[SS] “Both incorrect, but you’re getting warmer.”

[HH] “Eight-Seven suited”, Harriet the Hazy opted, like the next contestant on The Price is Right.

[SS] “Nope.”

[FF] “Seven-Six”, Figaro the Fish output.

[SS] “Still wrong.”

[TT] “Now it’s a cinch to derive / That it must be the Six-Five”, Tyrone the Telephone offered.

[SS] “Indeed. The Six-Five has just over 23% equity.1 But why is it slightly better than the Seven-Six?”

[TT] “The Seven-Six gets more hits / But the Six-Five gets more splits.”

[SS] “Indeed it does. Which ones?”

[RR] “Except for my initial guess, all these suited connectors can make the same number of winning straights and flushes, so that leaves full houses.”

[LL] “No, the Seven-Six makes more winning straights. They both win if there’s a wheel straight on the board, but a 65432 board is a win for the Seven-Six and only a push for the Six-Five.”

[EE] “So the 6-high straight is some of the Six-Five’s extra pushes. The rest are on the other side: the Jack-high and Ten-high straights, which don’t happen as often for the Seven-Six.”2

[SS] “And that’s most of the difference, although there are a few full houses. 77666 down to 77222 vs. 55444 down to 55222 accounts for a few more wins for the Seven-Six and a few more ties for the Five-Four. Lastly, JT98x wins more often for the Seven-Six than 432Ax does for the Six-Five.”

[SS] “The Six-Five has enough extra straight pushes to overcome the extra wins that the Seven-Six gets.”

[TT] “When facing Aces, your prospects are wee / Even with the best hole cards you could see / Out of thirteen tries, you’ll only win three.”3

Footnotes:

  1. The Six-Five suited has 23.056% equity, 0.0235% more than the Seven-Six suited (23.0325%).
  2. Out of 1,712,304 possible boards, the Six-Five suited wins 55 fewer times but ties 916 more times.
  3. This is an excellent estimate as three-thirteenths is the repeated decimal 0.230769.

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