Calculating EV

One thing I am struggling with is calculating EV. Can each one of you give me some examples of EV? It would really help. Also, any links that you like, too.

EV is not something you really calculate at the table. It's just too complex, as even the simplest of scenarios are rather lengthy. We'll look at this situation now so you can see EV in action.

Hero ($500)
Villain ($500)

Hero holds: J8

Board: ($100) 9 10 7 2 6 Villain bets $50


Okay, so at this point, we hold the absolute nuts. We know we want to raise, but how much? How about a min-raise that he can afford to call? Or what about a shove that maybe he'll interpret as a bluff? Or something right in the middle of the two perhaps? Some players are afraid of scaring off their opponnents, and habitually make tiny raises for value here, in an attempt to be called a high percentage of the time. However, the object is not to be called the most times (though it may help), the goal is to make the bet which on average returns the most money. To find which bet is best, we turn to expectation or "EV".

For this "easy" scenario, EV = (Probability opponent calls) * (Size of your bet/raise)

We will analyze three bet sizes, $50 min-raise, $150 raise, and $450 all-in raise. To finish the calculations we will need to approximate calling %'s for each of these bet sizes.

If you make the min-raise, our opponnent is very likely to call with most of his holdings, so we'll put the chances at 80%.
Betting $150 will likely fold most hands not containing an 8 for the non-nut straight, but the villain here is willing to fire at the obvious straight, so there's a good chance he does in fact have the 8. We'll guesstimate that he'll call the $150 bet 40% of the time.
Making the all-in move is quite strong, and again the opponent will probably not call without holding an 8. However, let us say large bets scare this villain, and there's only a 50% chance he will call you even when holding an 8. Therefore his likelyhood of calling the $450 is 20%.

To find out which is best we calculate the EV of each play.

$50 Raise
(.80) * ($50) = $40

$150 Raise
(.40) * ($150) = $60

$450 Raise
(.20) * ($450) = $90

In this case, the small "keep them in the hand" play is the worst of the three options, while the play which is hardly ever called is actually the best! Really in this case the only decisions were between making a small raise in an attempt to extract extra value from two-pair type holdings, or to say forget that I want to make as much as possible if this guy's actually got a real hand. Because there's still so much money behind, getting it in and hoping for an 8 turns out to be best.

You want to make sure you're making a bet in order to achieve the highest expectation possible, not because you want to get called, or because you want to out-level the guy into thinking you're on a draw when you really flopped the nuts. These factors may be the reason that a particular bet does have the highest expectation, but this is not the reason for making it. You are making the bet solely because it is the one with the highest EV.

As you can probably tell, it only gets harder from here. In other situations you need to factor in the likelyhood of your hand being best, being drawn out on, redrawing, etc., The list goes on.

I hope you take away from this, not the absurd math involved to figure out EV in more complex scenarios, but the basic concepts behind the reasoning to make this EV decision. Thinking in terms of EV can definitely assist in making you a better overall player.

Great post, Mase. A+.

As mace said, EV isn't something you bother to calculate since you are constrained by time. It is also essentially impossibile to accurately assigning %s to ranges or %s to the amount of times villain has a bluff as compared to a monster when you know you are probably dealing with a polarized range, etc.. The best you can do is estimate what line etc. will lead to the greatest profit.

So if you are in position playing a HU game vs an aggro player and are raised on the flop you are typically dealing with 1. air 2. a big hand like 2 pair/set 3. a draw 4. maybe a TP type hand. Lets say the board is 442r and you have KK. This is a board an aggro player is going to bluff raise alot because neither of you are likely to have much. There aren't any draws and villain simply isn't showing up with a big hand like trips here more than once in a million years. So range is basically air, or overpairs you beat that want to get it in on this flop. Lets say you figure that most of villain's range is air, then the most +EV play is to flat call the raise and to flat call turn and hope villain tries to make a huge 3 street bluff. You lose a huge amount of value by 3Bing this flop because villain's range is so weighted towards bluffs that can't continue. You are not worried about any scare cards or having to protect your equity, because even if an A shows up there isn't much chance at all that villain has one and you can easily call of your stack no matter the board if villain is likely to run a big bluff. So that is how you use EV in a general sense, without trying to calculate random $$ and % crap as to how to get the most out of your KK vs an aggro player on a 442r board. It is just a general term that deals with the value (positive or negative) you are likely to get out of a line given your assumptions about villain/game flow etc.

Great examples so far. I really appreciate it. Keep the examples coming!