Back to Basics: Thinking about your opponent's hand ranges

It's been awhile since I posted a "Back to Basics" segment, and was feeling motivated by Matt Tag's post on F*@! the math.  Therefore, I tried throwing this together.  I know this goes beyond a "basics" segment, but I was thinking about calculation of hand ranges anyway, so I wanted to get it out there.  Let me know what you think.

My chosen career, software engineering (systems engineering), was selected as a way to avoid using math directly in my head.  See, ironically, I'm maths challenged. Therefore, I try to avoid using math.  When I do, though, I am always looking for the "easy way out" - a shortcut or way to make the maths easier.

Figuring out equity, at any point, when faced with an all in situation, is somewhat easy, if you can do memorization and/or addition and multiplication.  We covered that in Back to Basics: Pot Odds, Hand Equity and Implied Odds, where you can count outs and multiply by either 2 or 4.  The more you do it, the quicker you'll become, as well.

However, figuring out your opponent's hand range is more difficult, because you have to deal with weightings and rough math; i.e. unless you know you're opponent's exact holdings, you cannot come to a precise number.  Let's go through the basics, first.  There are 6 hand combinations for any specific pocket pair (e.g. AA), 4 hand combinations of a specific suited combo (e.g. AKs), and 12 hand combinations of unsuited cards (e.g. AKo).  FYI: Surprisingly, it is statistically rarer to hold a specific suited combo (AKs) than it is to hold AA; you will hold AKs roughly 4/6 or 2/3 of the times than you will hold AA...  look it up in your HEM or PT3 databases (assuming you have enough of a sample size).

Okay.  We have covered hand combinations.  Let's talk about real-world examples:

Given that we now know there are as many combos of 2 pocket pairs as there are any combination of specific offsuit cards, AKo, for example, we can start to think about and assign ranges to our opponent.  We're in a PF situation, facing an all-in bet.  We hold QQ.  Regardless of how we arrived at the shove, let's assume that we know our opponent is capable of this play with precisely 4 hands: AKo, AKs, AA, and KK.  From the Back to Basics: Pot Odds, Hand Equity and Implied Odds post, we know that we have 20% PF equity vs. any overpairs to ours, but against 2 overcards, we have 50% equity PF.  So we assign a weight of 50% to the 16 combos of AK (4 for AKs + 12 for AKo) and 20% to the 12 combos of AA and KK: .5 * 16 + .2 * 12 = (.5 * 16 =)8 + (.2 *12=)~2.5 (figured out in my head by 2 * 12, move the decimal over one place) = 10.5.  Take that number divided by the total amount of combos (16 + 12 = 28)  is 10.5 / 28 = a little better than 35% equity in this hand (I took 10/30 = 33%, but the top number is bigger and the bottom number is smaller so I estimate around 35% equity) to make the call.  Therefore, we need a price better than 2:1 pot odds to make the call...  in other words, at best, we're splitting and at worst, we're dominated.  Given the range, this is an easy fold for most cases.

Let's move on to a post flop example:

Once again, we hold QQ and have seen a flop of 2 5 9 rainbow.  We're facing another all in situation and whether or not to call.  We hold a simple overpair and put our opponent on a fairly wide range of hands: 22-AA, AK, 87s.  I don't care what the PF action was, because I'm telling you what our opponent's range is precisely.  Do we really have to go through all the addition, multiplication and division to get to whether we're making a good call or not?  Well, let's think about it and do some shortcutting: 22, 55, 99, KK, AA have us dominated.  However, we're crushing 33, 44, 66, 77, 88, TT, JJ.  Therefore, we can roughly cancel out 5 pair combinations of domination / crushed because they are equal with equal weights (remember that all pocket pairs have the same number of combinations: 6 ), leaving us with 2 pair combos which we have crushed by an 84% margin.  This cancellation process gets us a lot further down the road, and more quickly.  What other hands are we dealing with here?  AKs, AKo, 87s.  There are 16 combos of AK @ 75% equity (remember that we have 75% equity on the flop vs. 2 overcards from the discussion on figuring out equities) and 4 combos of 87s at 68% equity (again, an open ended straight draw has roughly 32% equity).  The maths are as follows: 2 * 6 (the two remaining pair combos) *84% + 16 *75% (the AK combo) + 4 * 77% = [in my head] 12 * 80% + 16 divided in 4ths + 4* 75% = 9.5 + 12 + 3 = 20.5 out of 12 + 16 + 4 or 32.  Given the hand range we put him on, we are likely very far ahead of his range - roughly 66% equity to his 33%, thus any call will likely be profitable here.

What's that you say?  Your opponent isn't likely to be shoving 33, 44, 66 on this flop?  Or maybe he's likely to shove those types of hands on a bluff a certain percentage of the time?  Well, you can adjust your numbers and weights to play with the formula.  If you take out 33, 44, 66, the numerator increases and the denominator decreases, decreasing the likelihood that you are ahead.  Like I said, the world is not perfect; you will not be able to come up with an exact number that will tell you whether you're profitable or not if you make the play.  The idea is to come up with a rough notion of what to do.

You can certainly perform this type of evaluation in your head; the more you practice it, the better you will become.  There is a TON of further reading on the subject, if you care to get into the maths, as well as more detailed explanations.  I rather enjoyed Foucault's article on hand ranges, check it out if you're so inclined.  Finally, there is a tool out there available called PokerStove, which happens to be totally free.  You can compare equities on opponent's ranges vs. your holdings to figure out your equity in the hand.  It is a critical tool in your poker arsenal, if you're not already using it.

Remember: The overall goal of the exercise of putting your opponent on a range of hands is to determine that if you make this move in a vacuum 1 million times, will you be net profitable, net neutral or a net loser.  Take a few seconds, the next time you are facing an all in situation - or any action for that matter - and think about ranges and equities to the hand.  I am sure it will help you.

Comments welcome; please feel free to point out discrepancies.  I may not have explained the math perfectly - if you find problems, please let me know & I will correct them.  My goal at the end of the "Back to Basics" segment is to have a primer for which I can send friends and people I know  to better understand the game.