Calculating poker EV

EV, or "expected value", is the amount of money you can expect to win or lose on average in a given betting situation. In poker, if you are in a situation where your hand is the favorite and you win, your EV is typically lower than the actual amount you win because with EV, you also have to average in the times that your superior hand loses to draws that your opponent has. Obviously, if you are the favorite but lose the hand on a bad beat, your EV, or expected winnings, were much higher than the negative return you actually got on the hand, but since EV is an average, whatever probability you did have of losing the hand has to be figured into your calculation of whether or not you made a good bet in a given situation.

If your bet in a certain situation actually had negative EV, then you were making a bad bet, because on average this bet would lose money, but any bet with positive EV is a good bet because, on average, it makes you money. Similarly, bets with an EV very close to zero are simply marginal: they will neither make nor lose money in the long run. Since they do involve risks, though, in practice, they should generally be avoided unless there is some particular reason in your situation for taking a risk of marginal value. The measure for good and bad bets is not the winning or losing result of a particular hand. As anyone who has played poker at all is well aware, one does lose sometimes on good bets. But they are still good bets as long as they have positive EV. You may have lost the particular hand in question, but if that same situation, where you have positive EV, were repeated sufficiently many times, then your average winnings per hand would be very close to the expected value. Now, if EV is the measure for good and bad bets in poker, then it is obviously very useful to be able to actually calculate EV in a given situation. In NLHE, it is often difficult to do this with complete precision, but we already do know that we are particularly interested in distinguishing between bets that have negative EV (bad bets) and those that have positive EV (good bets). Moreover, one will also want to consider one’s own betting options in terms of whether one has a clearly higher expected value than another—and this can often also be done even when there are some uncertainties as to the exact "expected value" of one bet or another. Such uncertainties can be a result of unclarities regarding the range of hands your opponent might have given his betting up to that point or unclarities regarding how your opponent might respond to a given action on your part. As an introductory example for calculating EV, however, let’s consider a situation that comes up frequently in NLHE and with sufficient information to allow us to calculate EV rather easily.

Let’s say that you are holding As Qc in MP and that your opponent has Ad 3d in LP. There is only one card left to draw, and the board shows Qd 8d 3s 4h. There is $100 in the pot, and your opponent checks. You have $60 left in your stack, and your opponent has you covered. In this situation, you will win unless your opponent hits any of the remaining 9 diamonds or any of the 2 remaining threes, so your opponent has 11 outs to win the hand, and there are 44 cards left in the deck. In this situation, if you bet your remaining stack, your opponent does not have odds to call your bet—which isn’t to say he won’t… Players often do make bad calls, and if you have opponents that do so, then this will increase the amount you can win. But let’s consider this situation from the standpoint of your opponent. If he folds, his current EV is simply $0, which is always the EV of a fold. It doesn’t matter that he has already put money in the pot, since that money is no longer his. The money in the pot is up for grabs and the fact that he has stayed in the hand up to this point means only that he still has a chance at the pot, which he would win 11 times in 44 if you had just checked after him allowing him to see the river for free.

After you bet, however, he has the option of folding for no additional loss or calling your bet of $60. Can he beat an EV of 0 if he calls? Well, if he calls, he wins $160 ¼ of the time, but he loses $60 ¾ of the time. So, his EV is (1/4)*160 – (3/4)*60 = 40 – 45 = -5 So, by calling he on average loses $5 each time this hand is played, and he is thus better advised simply to fold. Now, what does the EV of this bet look like from your standpoint? We first have the problem of knowing whether your opponent will call or fold to your bet, but we can get around that to some extent by just considering these cases separately. If your opponent folds, then it is quite simple to calculate your EV, since you automatically win the pot. So, if your opponent folds, your EV is just $100. How about if your opponent calls? In this case, you lose your $60 ¼ of the time but win $160 ¾ of the time. So your EV in this case is (3/4)*160 – (1/4)*60 = 120 – 15 = 105 So, if he calls this bet, your EV is even better, as was to be expected, since your opponent was making a mistake in calling. And indeed, the additional $5 that he loses on average by calling is exactly the extra amount that you make over and above the $100 already in the pot. Hence, we see that by betting $60 here, you have an EV of at least $100, and if your opponent makes a mistake, it may actually be a little better than that.
Now, what would happen if you checked or bet a lesser amount? Well, if you check, then your opponent sees the river for free, so he wins the hand ¼ of the time, making at least $25 even if you fold to any river bet he might make. And now, you only win the hand ¾ of the time. So, if you check and just fold on the river when you’re beat, then your EV is now (3/4)*100 = 75 Your EV has now gone down by $25 by failing to bet. Now let’s assume you bet some amount x, which is less than $60, and your opponent calls. Your EV is now (3/4)*(100+x) – (1/4)*x = (1/2)*x + 75 We can see two things from this equation. First, as long as your opponent calls here, since you’re the favorite in the hand, the bigger your bet, the more you win. Any increase in x is always in your favor here. Moreover, in order for your EV to be any higher than the minimum of $100 that you had when your opponent folded to your $60 bet, then x has to be at least $50. If you bet $50, then your EV is just $100 whether your opponent calls or folds, hence a rational opponent will be completely indifferent to calling or folding to a bet of $50. But the advantage of making a bet bigger than $50 here is that it allows your opponent to make a mistake. If the probability is just 0 that he will call a larger bet of $60, then it really doesn’t matter, since you just take down the $100 pot every time. But if there is any probability at all that he will make a bad call (and there usually is), then this will increase your EV on the hand by $5 times the probability of a call.

Looking at this example from both sides, we can make the following conclusions regarding EV in NLHE: First, in order to determine EV with draws remaining, one must consider the probability that the underdog will win the hand by counting how many outs the underdog has to win the hand. Second, in order to protect a hand vulnerable to a draw, the favorite will decrease his EV, or average winnings, if he fails to place a bet that denies his opponent odds on a call. Third, folding a hand to a bet always has an EV of 0, hence calling a bet on a draw is only reasonable when the call actually gives you positive EV.

by Aisthesis