August 25, 2020

Counting combos to calculate rough equity

In the pot odds example you had the luxury of knowing exact which hands the villain was holding. Real poker ain't like that. You may, however, find yourself in situations where simple math can help you make the long term profitable decision, given the right assumptions.

Let's get specific - and still have the luxury of knowing what the opponent is playing, but only as a range of cards: You hold QQ and get 4-bet pot-sized in position (for instance on the button) from a player doing this with AK, AA, KK, QQ, JJ, TT and A5s. Is it profitable to call? Let's assume you get to realize your equity exactly against each hand (will only be exactly true if you have just that pot sized bet behind).

To answer if the call is profitable or not, we can use a range vs. hand equity calculator like Flopzilla and get the answer directly (and it is useful to know a few common match-ups by heart). But we can also use a more manual shorthand approach that is somewhat feasible at the table. We break down the opponents possible hands into:

- AA and KK, which have you crushed (80/20) - a total of 12 combinations, 6 for each pair. To count the combos just practice starting with Ah - which has three aces to combine with, then two more for Ad not yet considered, etc.

- AK (16 combos) and QQ (one combo) that you roughly flip with

- JJ/TT (6+6=12 combos) that you have crushed (80/20)

- A5s (4 combos) that you have dominated (almost 70/30)

In this example, we can easily see that the hands we crush and the hands we are crushed by cancel each other out in terms of combos. Since there is money in the pot - in fact the pot odds is 33% - and we get around 50% equity, we should clearly call, unless there is some specific reason to be risk adverse, like a tournament bubble. If your opponent sometimes includes 99 or AQ when doing this (as an exercise, count the combos of AQ - remembering you have two queens!), the call becomes even clearer.

If they never do this with QQ, JJ or A5s, suddenly we can see that we are up against roughly 40% (12/28 or 43% at 20% equity) combinations that crush us and 60% (16/28 or 57% at about 55% equity) that we flip with. Thus we can weight our equities in our head to a bit less than 40% by doing the 0.4*0.2+0.6*0.5 in our head. 

If doing arithmetic in your head is not your strongest side, a neat technique can be to visualize 20% and 55% in your head and go about 60% of the distance between them to find the estimate. The difference between 20 and 55 is 35, so you should go about 21 percentage units up, or to around 41% (which is in fact closer to the exact value than my rough math above where I substituted .5 for .55 to simplify the arithmetic). Still a call in our example, but getting closer - and the method and not the answer is the point here.

The same technique of counting combinations of bluffs (which busted flush draws can your opponent likely have given the action on the river?) and value can be used on the river, where you usually have much more information and also more known cards. Some of the known cards are in your hand, which is commonly referred to as blockers - which is the topic of our next small piece.

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