Poker Math: Extending the 2/4 Rule

Three more math books arrived today, now I have a stack of 9 to get through, maybe 1-2 years worth if I learn fast. That means even less hope for poker, so I might as well pass on another idea I’ve had that I probably won’t have time to work on.

If you guys don’t know what the 2/4 Rule is, it’s pretty simple. Take the number of outs you have and multiply by 2 to get the approximate percentage of hitting it with 1 card to come, multiply by 4 with 2 cards to come. For example: you hold a flush draw (9 outs), 2*9=18 so you have about 18% to hit it on the next card, or 4*9=36% to hit it on either turn or river. The real odds are 19% and 35% respectively, but this negligible error is small price to pay for fast estimates. And you don’t have to memorize a table.

What interested me most about this is that after some training, I figured it may be possible to predict your equity vs a range with decent accuracy. This would take some work and wouldn’t be as useful for situations where decisions will have to be made on future streets, but for AI situations I thought this might work. For example, if on the flop your opponent has two possible hands, equally probable, and you have 6 outs against one and 8 against the other, then you don’t need to convert to percentages, you can just average them first and then apply the 2/4 rule. In this case, you’d have 7 effective outs and so ~28% to win with 2 cards to come. Another nice benefit is you can work with fractional outs, whereas if you just memorize the chart you get stuck with a harder, less accurate approximation.

Below are some flop matchups. This assumes 2 cards to come, with leftmost hand being the underdog and it’s your holding. What I did was run the hands through poker stove and then just divided by 4 to get the odds-equivalent to plug into the 2/4 rule:

  • Overpair vs 2-pair: 6 outs
  • FD vs Set: 6 outs
  • OESD vs Set: 6 outs
  • GS+FD vs Set: 9 outs
  • OESFD vs Set: 11 outs (same for OESFD without a straight-flush draw)
  • Set vs Flush: 9 outs
  • Set vs Straight: 9 outs
  • Top 2-pair vs Any Set: 3 outs
  • Top/bottom 2-pair vs Any Set: 2 outs

You’d have to memorize these and their converses if you wanted to apply them because subtraction would just add needless complexity.

A couple of interesting backdoors:

  • Backdoor FD (3-card flush): 1 out
  • Outside Straight (have 6 7 8): 1 out
  • Inside-Outside Strt (have 6 7 9): .7 of out
  • Inside-Only Strt(have 5 6 9 or 5 7 9): .35 of out

I think it’s ok to just use 1 out for 3 card straights and half of an out for any sort of inside runners. You can also couple the backdoor FD with any of the three possible runner straights (but you can’t have two backdoor straight draws), so you can have up to 2 outs worth of these backdoors. Tilt Equity anyone?

Also, it would be beneficial to know exactly how many combinations of each hand there are if you intend to estimate ranges (you should know most of these if you read Harrington’s books):

  • Unsuited Unpaired Card Combinations (ex AKo): 12
  • Suited Unpaired Card Combinations (ex AKs): 4
  • Total Unpaired Card Combinations (ex AKo/s): 16
  • Pair Combinations (ex AA): 6 (4 if one of the cards is dead/in your hand, 1 if two cards are dead/in your hand)
  • Possible Axs with 2-flush board: 10 (ie, how many Axs are possible for your opponent to hold if there’s a 2-flush on board)
  • Single 2-pair combinations: 9 (ie, all JTo/s holdings on J T 5 board)
  • Single pair combinations: 12 (ie, all AKo/s holdings on K 9 6 board)

A basic outline of the equity-calculating training:

  1. Practice seeing what possible holdings there are for opponents. You should be doing this just for standard hand analysis anyway.
  2. Practice calculating combinations of each (card removal plays a big role in possible hand combinations).
  3. Practice your matchup in odds against each group. I suspect at this point you’ll start to “see” hands with similar odds as one group.
  4. Practice calculating the weighted average for your odds vs the range. With enough practice, I think this will become pretty natural and eventually you’ll just sort of sense the odds.

I don’t expect many people to be interested in this. It still requires a lot more work to make useful, and a lot of practice to get the system down, but poker isn’t supposed to be easy. I think this system would be very useful to a live player, since if you’re playing 8+ tables online, you really don’t have time to think through the calculations. Nevertheless, I think at least studying these numbers will help develop a certain “number sense” for poker equities. I’m not sure how many players can actually estimate equity vs ranges with a good degree of accuracy, I expect most successful players just do it through countless hours of table experience and hand analysis.

Have fun!

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3 Responses

  1. lol i hate maths but what you write makes sense.

  2. That’s some deep shit D.

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