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:

- Practice seeing what possible holdings there are for opponents. You should be doing this just for standard hand analysis anyway.
- Practice calculating combinations of each (card removal plays a big role in possible hand combinations).
- 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.
- 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!

]]>This summer I did a little analysis of required fold equity (FE). I had high hopes for the project, but nothing great came of it. What did come out of it, however, was a chart that might be of use to you guys. It’s not what I wanted to release, but I don’t really have the time to do more meaningful analysis atm.

The chart assumes the pot is unopened when you’re to act (either you’re first in or it’s checked to you). So, for example, if you’re thinking of *raising *with FE this chart wouldn’t be accurate and you’d need a little more fold equity. It also assumes your opponet will either call or fold and there won’t be any further action after. If your hand has implied odds (ie, you’re drawing to best hand) then you stand to win a little when you hit and so can have a bit less FE. But, keep in mind your opponent can potentially raise and possibly deny you the win%. If you have reverse implied odds (ie, much of your win % includes making a pair that could give someone 2-pr) and you stand to potentially lose a little if you make your hand, you want a bit more FE.

Also, situations with more than one opponent get trickier and I wouldn’t use this chart for that. For example, if each opponent will fold 40% of the time, your FE is only 16% (you want both to fold, so .4*.4). So technically here you’d have to find the square root of the fold percentage and that’ll give you the required average FE of each opponent, but even then the implied odds get messy.

Nevertheless, this chart made clear to me that what I considered required fold equity was way too conservative. For example: betting 2/3 of the pot with just 10% to win drops the required fold equity from 40% to 30%, which is a 25% decrease.

I thought it was pretty cool how small sources of equity can combine to make for an unexpectedly profitable situation. My goal as a poker player was to identify hidden sources of equity, but it was cut short when I decided to pursue math and school.

Related post: Required Fold Equity equation

]]>I made one for Full Tilt 6-max tables that closely resembles the PA HUD-type layout, but with the powerful new 3-bet stats of HM. If you want to use it for other sites, you’re going to have to tweak the stat positioning a bit to match it up to different seat layouts.

**Pop-ups**

These show up when you move the mouse over appropriate stats.

You can download it here: http://dwarrior.110mb.com/poker/DWarrior-HM-HUD-1-1.zip

Unzip this into the Config folder of your HM directory (by default, C:\Program Files\RVG Software\Holdem Manager\Config)

I’m putting it up so people can start using it and give me feedback as to how I can improve it, let’s create a kick-ass HUD!

Note: I really wish HM provided a “Blind vs Steal” stat, which combined SB and BB. For now, I chose to display SB only because BB is somewhat skewed by Battle of Blinds situations. I chose to move it off to Steal pop-up for when it folds around to your SB and you want to check how the BB defends.

Note: To manually align the stats, press and hold Ctrl+Right Click and drag.

]]>EV = our expected value

f = probability our opponent folds to our bet

p = current pot (before our bet)

*Note: I’m completely ignoring any further play*

EV = (fold equity) + (SD equity given call)

EV = f*p + (1-f)*(SD equity)

EV = f*(p – (SD equity)) + (SD equity)

We of course want EV > 0, so

f*(p – (SD equity)) + (SD equity) > 0

f > -(SD equity)/(p – (SD equity))

(SD equity)/(SD equity – p)” width=”147″ height=”41″ />

To make this equation complete, let

w = probability of winning at showdown when called

b = our bet

then,

These equations make sense because:

- When we have 0% chance to win if called, our showdown equity is –b, so we have:

*f > b/(b+p)* - On the other hand, when we have a lock (100% chance to win), our equity is p+b (the pot plus opponent’s call), so we have:

*f > (p+b)/(p+b-p) = (p+b)/b*, which means we never want the villain to fold - This also tells us that the result “f > 1” lies somewhere between 0% and 100% to win. This is obviously because of pot odds, since we win
*(p+b)*, but lose only*b*, we don’t actually need a lock to want the villain to call every time (ex: when you have straight vs set, you never want opponent to fold, even though a set will suck out quite often)

Related Post: Chart of Required Fold Equity

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