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In a game of heads up Texas Hold'em, what hand has the highest odds of winning, while still holding the worst hand on the flop? For example, if I have KsJs, my opponent has 3d2h, and the flop comes Qs10s3c, I have the worst hand, but with 21 outs to the turn and river, I have roughly a 70% chance of winning the hand at showdown (any ace, king, jack, nine or spade puts me ahead).

I've messed around with a number of other hands, but it seems that the flush and straight draws with over cards to the opponent's pair gives the most outs. Are there any hands/flops with better odds of winning while still being behind?

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  • 78hh vs 2s2c on 6h5h3s is 72.3% to win.
    – Andrew Chin
    Jul 6, 2022 at 13:44

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You've maximised the number of outs the temporarily-weaker hand can have to improve to a made hand on its own merits: all 6 outs to a higher pair, and all the straight and flush outs. However, there are a couple of subtle details that can improve the odds.

The first concern is the opponent's redraw chances. As it stands you could get a king or jack on the turn, but the the river could bring a 3 or a 2 putting the opponent ahead again (Or river and turn vice versa) giving opponent five redraw outs against your higher pairs. This chance is minimised if you change opponent's hand to the pocket pair 2d2h (just two redraw outs against your higher pairs).

This also means that you no longer need 3c in your flop, so we can ask if there is anything else we can do with the third card. The question is really whether it is better for us if the board is paired: say Qs10s10c. This gives us three more outs to counterfeit opponent's pocket pair with another queen, plus the miracle chance of both remaining tens coming out. But it also changes the redraw chances: the remaining 2's are redraws to a full house so they beat not just our higher pairs but our straights and flushes as well (though not our two straight flush outs ;-)).

(Edit: I'm not convinced anymore that the paired board is better. I also missed the chances that the straight or flush comes but a 10 gives opp a full house.)

My first back-of-the-envelope estimation of the change in chances is:

  • Three more outs after the flop is approximately a 12% higher chance.
  • We have six straight outs and seven flush outs potentially hurt by the new redraws: I'll overestimate that to about 52% win chance potentially affected. The opponent needs to hit one of their two outs on the single other card: about 4% chance. So the combined chance of these redraws is just over 2%.

Even with the overestimate, the extra opposing redraws should be much less than the the benefit of our extra outs. So the paired board ought to be better for us. (Though I'd want to put it through a simulator to be certain.)

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  • Very nice answer, and good catch on the redraws! Putting in the hand/flop into an odds calculator gives a 71% chance of winning, so 1% better than my example
    – caird coinheringaahing
    Jul 6, 2022 at 1:42
  • I just tried messing around with the odds calculator myself, and I'm less sure now that the paired board is better.
    – Chessanator
    Jul 6, 2022 at 1:47

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