Hand Equity v Random Hand

I’ve just finished playing a bit at 1/2, which got me thinking about what hands I might shove with if I was last to act, and heads up against someone that had shoved pre-flop 10 or 20 hands in a row (suggesting that they were probably shoving with a 100% range).

Here are the best hands against a random hand, with the percent equity for each. Note that this only applies heads up. Even if you had 2 players going all in every hand, the rankings will be different when you are up against 2 ranges (small pocket pairs, for example, will do worse). Note also that this ranking will also change quite a bit against smaller ranges that don’t consist of every possible holding.

I found it surprising that AKo does not make the top ten in this particular situation, and that AJs actually ranks just above it. I also find it a bit interesting to see how much higher 55 ranks than 44, 33 and 22.

Some tidbits and surprises:

  • I’d already known that K2 and every king beats a random hand
  • KQo is not in the top 20
  • Wow, J8o and J6s???
  • Q6o and even Q5o?
  • T9o appears to be the lowest, unsuited hand
  • T7s and 98s the lowest suited hands (note that hands like 76s will actually do quite a bit better multi-way, as other hands will often block each other)
  1. AA 85.2
  2. KK 82.4
  3. QQ 79.9
  4. JJ 77.5
  5. TT 75.0
  6. 99 72.1
  7. 88 69.2
  8. AKs 67.0
  9. 77 66.2
  10. AQs 66.2
  11. AJs 65.4
  12. AKo 65.3
  13. ATs 64.6
  14. AQo 64.4
  15. AJo 63.6
  16. KQs 63.4
  17. 66 63.3
  18. A9s 62.8
  19. ATo 62.7
  20. KJs 62.6
    A8s 62.0
    KTs 61.8
    KQo 61.5
    A7s 61.0
    A9o 60.8
    KJo 60.6
    55 60.3
    QJs 60.3
    K9s 60.0
    A5s 59.9
    A6s 59.9
    A8o 59.9
    KTo 59.7
    QTs 59.5
    A4s 59.0
    A7o 58.8
    K8s 58.3
    A3s 58.2
    QJo 58.1
    K9o 57.8
    A5o 57.7
    A6o 57.7
    Q9s 57.7
    K7s 57.53
    JTs 57.5
    A2s 57.4
    QTo 57.3
    44 57.0
    A4o 56.7
    K6s 56.6
    K8o 56.0
    Q8s 56.0
    A3o 55.8
    K5s 55.8
    J9s 55.7
    Q9o 55.4
    JTo 55.2
    K7o 55.2
    A2o 54.9
    K4s 54.9
    Q7s 54.3
    K6o 54.2
    K3s 54.1
    T9s 54.0
    J8s 54.0
    33 53.7
    Q6s 53.6
    Q8o 53.6
    K5o 53.3
    J9o 53.3
    K2s 53.2
    Q5s 52.8
    T8s 52.3
    K4o 52.3
    J7s 52.3
    Q4s 51.9
    Q7o 51.8
    T9o 51.5
    J8o 51.5
    K3o 51.4
    Q6o 51.0
    Q3s 51.0
    98s 50.8
    T7s 50.6
    J6s 50.6
    K2o 50.5
    22 50.3
    Q2s 50.2
    Q5o 50.1

That’s it. Every other hand loses to even a random hand.

1 Like

Oh, here is the bottom 10. Note that many of these are connected… straight potential not helping that much (strange that straights don’t beat flushes, given that they are rarer).

73o 36.6
53o 36.3
63o 36.1
32s 36.0
43o 35.1
72o 34.6
52o 34.3
62o 34.1
42o 33.2
32o 32.3

That does seem rather odd. The average card is an 8, because there are 7 values lower and 7 higher. I would expect hands containing any 10 to do better against a true “any 2 cards” range.

I suspect domination is a major factor in determining overall performance. 76s performs reasonably OK against a mix of AK, JTs and 55, but throw in T6, 96 and 86 and the picture changes quite rapidly.

Here’s a fun matchup (cheating a little to try and give 76s a really big advantage by deliberately creating interference with the bigger hands).


1 Like

Yes, I have recently been trying to revamp my tournament game a bit based in this kind of information.

Yes, 55 does better than you might expect, but then every straight requires either a 5 or T, so 55 has a powerful blocking effect especially on the pairs 22, 33, 44, but also to some extent on 6 thru 9, and in a number of cases where 22, 33, 44, 66, 77, 88, 99 do make the straight, 55 will make a boat (or quads) if the board pairs. So with 55 vs 44 on a board of A23, when the turn is a 5, then 55 will still have 10 outs on the river plus 3 more outs to chop the pot, so 44 will not win 25% of its straights.

One thing I have tried to eliminate from my play is calling raises with dominated hands like KTo and KJo, as they are not as good as they look since early raisers are likely to have the dominant hands in their range.

On the other hand, non dominated hands can make a lot of sense for a call if you are in the blinds, for example with early raise from UTG, flat call from Button.

Here you certainly expect that at least one hand, and quite probably both have two high cards, for example UTG has AJs and Button has AQ, and you might reraise with 98s, knock UTG out before the flop, then bet if the flop has no Ace and no more than picture card. If these players both call, there is a good chance that they will block each others aces, and if they both hold AK, the effect is even more pronounced. So if the flop comes T 5 6 with one card of your suit, I would definitely want to take on two face cards.

The beauty of this kind of call or raise with a nondominated hand is that the opponents may be blocking each other, but also that you have almost the same betting advantage as the button, in the sense that you can see the flop and make a decision to throw away your hand immediately if it whiffs completely, or if you do like the flop then you can lead out and try to fold out your opponents before they have any chance to bluff from late position.

However, sometimes it will all go wrong anyway, no matter how right you think you are. See below.

In this hand large stack limps in, button calls, and I have AA so I raise to 4BB from BB with the intention of folding out the two limpers (should have raised higher), but UTG calls. Flop come 886. I put the opponent on either an inferior A like AJ or AT, or more likely a small to medium pocket pair give the limp-call. I shove the flop so that he cannot call any draws, and he turns over A8s, having flopped trips. What do you think of his preflop call? I don’t think I would have called in his place, but maybe he was playing 4-dimensional poker.

Of course if I had raised higher preflop, he might have folded, but I wanted a caller with an inferior hand, and anyway the slider on the RP interface is hard to operate accurately with a mouse, and you will often end up wildly overbetting or timing out while trying to get it right, so it is easier to bet the pot or half pot with the button provided. It would be really useful to have some other presets for 1/3 pot, 3/2 pot, and double pot.

This was annoying as I was well placed to win this tournament, but it was not to be. Even if I had not shoved the flop this hand would have been very hard to get away from without taking heavy losses.


Shove ranges should widen when you have less than 10 bb … so even if you are 50% pf … you may get a fold from one who should call …fold equity!.. in the highlighted hand … w/ ~ 30 bbs each …you made an excellent OOP raise … pricing your opponent into a bad call … the flopped trips was a 6%er from ML and is just a cooler … rarely will you ever fold AA in that spot … now from ML’s perspective … ML called another 1800 to win 3600 so about 33% of a 5400 pot … she has an A blocker … you are 4 handed … she’s suited… she’s in position … and if you are playing correctly … you have some KQ type hands in your pf raise range where she is ahead… soooo ML being a tough player makes the call … and IMO she can expect to have a 33%+ chance at winning the hand w/ A8s … so i would say well played by both of you