To find out if I'm doing EV calculations correctly or not, I am going through problems that I can find and then comparing my answers with the ones that are given. The example problem given in this post is about the one in Collin Moshman's "Head's-Up No Limit Holdem" book, starting on pp8. It reads as:

"You hold the 5s,4s and the flop is Ad,7c,6h. Your opponent goes all-in for $100 in a $400 pot. You believe he has a pair of aces, so what is your EV if you call?" Below are two of my attempts at solving this problem. Below those are my conclusions and some second-guessing when it comes to the two methods that I used to arrive at the equity figures used in the formula (%frequency call):


NOTE TO SELF - DOES THIS FORMULA WORK HERE (using Equilab to get equities)?:
EV(call) = [pot+villain'sbet]*[%frequency call and win] - [bet]*[%frequency call and lose]
EV = [$400 + $100]*[0.26] - [$100]*[0.74]
EV = ($130) - ($74)
EV = $56

NOTE TO SELF - DOES THIS FORMULA WORK HERE (using odds/outs/rule2-4 to get equities)?:
EV(call) = [pot+villain'sbet]*[%frequency call and win] - [bet]*[%frequency call and lose]
EV = [$400 + $100]*[0.32] - [$100]*[0.68]
EV = ($160) - ($68)
EV = $92 (CORRECT - This is what the book gives as the method and the answer...)

So, I guess that the EV(call) formula above does in fact work for calling an All-In bet on the flop. However, when estimating equities for this formula (%32 and %68) by counting odds and outs and then using the rule of 2 and 4, it produces a resulting EV that I suspect could be a far different answer from the answer I get when I use Equilab to calculate the win and loss frequencies for the formula. I have no doubt that the book is correct in its math, and I'd also venture to say that using an equity calculator (Equilab, Stove, etc) probably produces more precise results.


My questions are:
1) when doing hand history reviews, shouldn't I just go ahead and use Equilab for calculating precise frequencies, rather than using estimation? After all, I'd be reviewing the hand histories away from the table anyway, so I'd have the time available to get more accurate results.

2) Or, maybe it's just a case of whichever method I use to get equity results for the EV formula, using either an equity calculator or estimating equity using odds/outs/rule2-4; they are both accurate enough to use as long as I stick with the same calculation method when comparing various lines of play?

3) In other words, since estimating equity using odds/outs/rule2-4 is much faster, maybe I should go ahead and use this method where possible (but for each related play comparison), and only choose to use an equity calculator for instances where it's needed (but also for each related play comparison)?


It's probably not as big of an issue as this post seems to make it out to be. I just don't want to be all confident in my methods and then set out reviewing hand histories and make bad assumptions of whether a line of play is good or bad because my method of calculating EV is wrong. I would appreciate hearing any thoughts on the above 3 questions. - Thanks

When the book says "you believe he has a pair of aces", it doesn't mean he has AA. That would be "you believe he has a set of aces".

54ss vs { A2o+, A2s+ } on this flop is 0.3162, so about dead on.

did it with equilab and got the same results as with the 2 and 4 Rule

It seems your questions focus on whether you should use outs or an equity calculator to do an off-line EV analysis. The equity calculator will take longer but give you ‘exact’ results (with minor simulation variability if that is what the simulator uses).

The outs method is faster but subject to two types of errors: errors in counting your outs and errors in the 2x and 4x rules. The latter are relatively small ( there are more exact rules, e.g. Solomon’s rule) but odds counting is more critical and can sometimes be a bit tricky for opponent outs can ‘counterfeit’ your outs.

If you’re a stickler for accuracy use the calculator. If you use outs, occasionally check your counting accuracy with the calculator. The advantage in using outs off-line is the experience you gain, which can then allow you in many cases to make reasonable equity estimates while playing.

Thank you all for your replies. It seems that I was using the wrong input for villain's range (I had used AA instead of Ao+, As+).

@statmanhal -- "It seems your questions focus on whether you should use outs or an equity calculator to do an off-line EV analysis."

Yep, that's exactly the way I would've asked it if I had been able to put it 'to the point' like that. Feels kind've silly now - all my concentration was used up by some relatively simple math, so that I couldn't concentrate on much else. I think it's the word problem aspect of the whole thing that makes my ears steam once in awhile.

"The equity calculator will take longer but give you ‘exact’ results (with minor simulation variability if that is what the simulator uses). The outs method is faster but subject to two types of errors: errors in counting your outs and errors in the 2x and 4x rules."

Thanks for that advice. I think I'm going to do something similar to what you say. I'll probably start off by first calculating the EV by counting outs, to make sure that I'll be able to catch it if I make any glaring mistakes at it. Then I'll go ahead and double check my math with a calculator to see whether or not I goofed when doing it by counting outs. I had'nt thought much of that possibility, but after abusing my brain a bit when learning these EV calculations in various contexts, I can definitely foresee a time when I'm liable to make some huge mistakes when counting outs, due to most of my attention being elsewhere. However, thanks again statmanhal, I think I'm able to go ahead and settle on using the above EV calculation for situations where I want to know the value of calling an All-In bet on the flop. Now I'm going to see if maybe I can also use the same formula for preflop calls (probably not) and then turn and river calls (maybe). Then, once I get those down good, I have another formula for the EV of betting that I need to gain some confidence in my ability to apply it.


Thanks for catching that mistake, ChrisV. I had noodled around a bit more with the calculations and some thoughts that I had on the pros and cons of using odds/outs vs an equity calculator for EV calculations, and I was going to include that stuff in a follow up post to the OP, but since you caught a mistake I had made, the equity results are much closer now and I don't have to worry about the differences as much.


@ thoughtsymmetry - Yeah, I get nearly the same results with both methods now also. The way I had mis-read Moshman's example in the OP, I thought that it had intended for villain to showdown with AA (which I guess it did). But I goofed by using a range for villain of AA, in Equilab, instead of Ao+, As+.

I tried to set up a decision tree of this scenario in the trial of CREV (Hero is the button with 5s'4s; villain is the BB with Ao+,As+; the flop is Ad,7c,6h; villain goes All-In for $100 into a pot of $400 and Hero calls). Apparently, I'm not setting up the decision tree correctly, because CREV gives a resulting EV for Hero's call of EV = -33.15. However, the equities that it gives both Hero and villain before showdown are within %1 of what I get when doing the calculation by hand - they are %33 and %67.

Being new to the software and still learning it on the side, while/until I learn the EV calculations by hand first, I setup various types of decision trees trying to find out where I was making a mistake but I kept arriving at the same results, an EV for Hero's call of -33.15.

Does anyone who uses CREV enough and also happens to read this post have any idea of what I might be doing wrong when I'm setting up the decision tree for this hand? I posted a picture of the decision tree at http://imgur.com/OKGUx9z

Just after I submitted that last post, I re-read the scenario and realized that I did not adjust the pot size in CREV to be $400 when villain goes All-In for $100, so I'll bet that's probably where the difference in EV results comes from. I'm going to play around with the program some more and see if I can fix the decision tree...

I think you just need to add $400 dead money to the pot before villain shoves (you can add dead money by clicking on the 'single chip' icon, where you can also set the size of the blinds, rake etc. There's an 'Add chips to pot' option at the bottom of that menu.)



So EV(call) of $89.69 instead of $92 due to Hero's equity of 31.6% instead of 32%.

Thanks DiamondDog. Before seeing the image of your decision tree, I used my last decision tree (linked above) and added $400 of dead money to the pot. I also changed the conditions of the BB's holdings when going All-In from ">=tp" to "all hands", like your image shows. But the closest that I could get the EV results to $92 was $100.55. That's not a whole lot off, I guess, but after seeing that your decison tree produced EV results that were only off by about $2.30, I looked at the one that I was using again. I couldn't track down what was wrong with mine, so I just started a new tree and did everything from scratch. That worked and I got the exact same results as in your decision tree.

I also downloaded your EV spreadsheet a couple of days ago. That's quite a handy resource to have, along with all of your "Mathenoobics" articles. As I confirm that my math is correct for various EV formulas, I'm then putting each working one into a spreadsheet so that all I have to do is input several values to get EV results. Hopefully, by the time that I get the spreadsheet finished, then I'll be able to use CREV well enough that I don't make many mistakes when setting up decision trees. I'll probably end up using the spreadsheet for evaluating the EV of relatively simple lines of play, and then get CREV so that I can quickly evaluate the hands that have more than 5 nodes to them.