Yesterday I went and determined the amount of fold equity we needed in order for a bet/3bet all in to be break even with AcJc on a Ad5c4c board. If you remember, I found that the break even fold equity for that hand to be 62.0 percent. Well, how do we know exactly how much fold equity we have?
It wasn’t long ago that I thought this number was imaginary, or something that was just guestimated with somewhat accuracy and numbers thrown out there by the ‘in-the-know.’ Not true at all! Growing up in limit hold’em I never really had to deal with the concept of fold equity but mostly just trying to find ways to get value out of opponents. However, my limit hold’em days were years ago and now its no longer an excuse.
To start with fold equity, we have to put our opponent on a range. This can either be concrete or variable but you can’t determine fold equity without at least putting your opponents on a range of hands. In our example, you can either start with the hands that your opponent is raising with on the flop or just calling the all in. I prefer to do the easiest one first, which happens to be what is he calling an all in with? We determined yesterday that our opponent was only calling an all in with a set: 55, 66, or AA.
Now that we have his calling all in range, we have to find out how many total combination of hands he can possibly have with the set. Given the board of Ad5c6c and our hand of AcJc, our opponent can only have 3 combinations of pocket 5s (5d5s, 5d5h, 5h5s), three combinations of pocket 6s (6d6s, 6d6h, 6h6s) and only one combination of pocket paces (AsAh) for a total of 7 combinations.
After we have the number of calling combinations, we have to find the number of raising combinations; in other we words lets find out his raising range on this flop. We already know he’s raising 55, 66, and AA but what else can he raise? To find this let’s first find out what his preflop raise (PFR) range is: the read given was that he raised 8% of his hands, but that number isn’t exactly accurate.
I say that the number isn’t accurate is because the 8% PFR is a weighted average over all of our opponent’s raising range. The most accurate number would be his open raise % given each position, a number not available in Poker Tracker but is available in Hold’em Manager. I can guestimate that our opponent’s open raise range in Middle Position would probably be something like 10-13%. If we use 13% as a figure, that puts our opponent’s range somewhere around:
44+,ATs+,KTs+,QJs,JTs,T9s,98s,87s,AJo+,KJo+
If you were to plug 13% into Poker Stove you wouldn’t get this exact range, and that’s because our opponent isn’t always opening the exact top 13% of his hands, but a sometime polar range including smaller pairs and suited connectors. However, we can look at that opening range to determine what he is going to raise us with on the flop. If you take out pure bluffs and include only semibluffs, his semibluffing range here would probably be: 8c7c, KcQc and maybe KcTc. That increases the total number of combinations we have from 7 to 10, but even if he folds the two K high flush draws if we shove we still don’t have the required 62% fold equity. In fact, we don’t want him to fold his two K high flush draws so shoving would be pretty horrible there.
What we’re wanting him to fold though are his value raising range, maybe AQ but especially AK. Again, some players wont even raise a hand like AK here and most wont raise AQ unless they have a read or are willing to stack off with it. Say our opponent raise/folds AK in this situation, that gives him a possible 8 possible AK combinations that he could fold after a raise (and also 8 AQ combinations if you want to put that in there).
Okay, we have the number of raising combinations:
Sets: 7 Combos
Top Pair: 8 (16 if you include AQ)
Semibluffs: Max 3 (none of which we actually want to fold if we shove)
Total raising combinations: 15 (23 with AQ, 26 with semibluffs).
We already determined he was only going to call us with sets only, so that gives him 7 calling combinations out of 15, or 8 folding combinations. Our Fold Equity then with that range is 8/15, or 53.3%, which is clearly below our Break Even number of 62.0%. What if he raises AQ and folds it? Well, now that is 16 folding combinations out of 23, so our fold equity there is 16/23 or 69.6%.
So, in order for our shove to be profitable our opponent has to raise/fold AQ and AK in this spot. How profitable? Well, all we have to do is plug in our two percentages back in the EV Formula from yesterday:
EV = %Fold * Value of fold + %Call*Value of a call
Where:
% Fold = 53.3% and then 69.6%
% Call = 46.9% and then 30.4%
Value of fold = $106
Value of Call = -$173.03
Plugging these numbers in, you’ll find that the EV when our opponent raise/folds either AK OR AQ is:
.533*106 + .469*(-173.03) = 56.50 - 81.15 = -$24.65
And the EV when our opponent raise/folds both AK AND AQ is:
.696*106 + .304*(-173.03) = 73.78 - 52.60 = +$21.28
Thus, you can see there is a significant difference between the EV of the two scenarios, where if he raise/folds AQ then shoving is +EV and probably superior than just bet/calling or check/calling. But, this isn’t the place for that exercise. However I’ve specifically proven when it is correct to shove our hand and when it is incorrect to:
It depends on our opponent’s raise/folding range in this situation and our equity when called vs that range..
