August 14, 2009
The first tool used to analyze a betting decision is the concept of expected value (or EV). In order to calculate EV, you make a list of all the possible ways the hand can play out; for each possible scenario, you multiply the number of bets you will win or lose in the given situation by the probability of that situation occurring, and then you add up these numbers for all the possible scenarios. A concrete example will hopefully make this clear.Example 1:
Suppose that a short-stacked early position player raises with Ad Ks; everyone folds to you in the BB and you call with 8h 7h. For the purposes of this calculation, we will assume that the folded SB is equal to the rake paid in the hand, so that the pot has exactly 2 big bets in it at this point. The flop comes Ac 9d 6s. You check, your opponent bets, and you call, so there are now 3 big bets in the pot. The turn is the 9h, you check, and your opponent makes a full bet which puts him all-in. There are now 4 big bets in the pot, and the question is whether you are better off folding or calling at this point. This can be determined by comparing the EV of the two options. The EV of folding is easy to calculate: there is only one scenario where you gain 0 bets and it occurs 100% of the time, so the EV of folding is 0. The EV of calling is only slightly more complicated. There are now two possible scenarios: either you hit one of your 8 straight outs to win the pot or you miss your straight and lose the pot. Letâ€™s calculate the probability of these two events first. There are 44 cards remaining in the deck (52 minus the 8 cards accounted for on the board and in the two playersâ€™ hands). Since each card in the deck is as likely as any other to be the river card, the probability of making a straight is 8/44 = 0.182. The probability that you donâ€™t win can be computed in two ways: either as 36/44 = 0.818 (since 44 â€“ 8 = 36 cards cause you to lose the pot) or simply as 1 â€“ 0.18 = 0.82 since the sum of the probabilities of all possible events must add up to exactly 1. If we call and make our straight on the river, we win the 4 bets that are in the pot, while if we call and lose the pot, we lose the 1 bet that we used to make the call of our opponentâ€™s turn bet. (When calculating the EV of the turn decision, we donâ€™t consider how much money in the pot came from our previous bets, as we no longer have control over those bets at this point in the hand. Our decision is the same no matter how the 4 bets ended up in the pot on the turn.) So to determine the EV of calling, we compute (amount won when we make our straight) x (probability of making a straight) + (amount won when we donâ€™t make our straight) x (probability of not making a straight) = (4 bets) x (0.182) + (-1 bets) x (0.818) = -0.09 bets. We interpret this calculation by saying that, on average, calling the turn bet loses 0.09 bets. In general, we write down all our formulas in terms of how much we â€œwin,â€ subject to the understanding that winning a negative number of bets is equivalent to losing the corresponding number of bets. In this admittedly simplified and somewhat contrived example, the better option is to fold.Example 2:
Suppose that we keep the action the same as in the first example, but give our short-stacked player two additional big bets. Thus, when he bets the turn, he still has 2 big bets left in his stack. We will further assume that if we call the turn, he will bet any river card and call a raise no matter what card comes. Let us now consider whether folding or calling is better on the turn. (Hopefully, given our assumption that our opponent never folds his hand, you see that raising the turn is clearly worst, though itâ€™s a good exercise to go ahead and compute the exact EV of raising the turn if you assume either that your opponent will reraise you all-in or if you assume that your opponent will call the turn raise and call any river bet.) As before, the EV of folding is still 0. When calculating the EV of calling the turn bet, we must make some assumptions on how we will play the river. Let us suppose that we will fold the river if we do not make a straight (that includes folding if we make a pair of 8s or 7s on the end), and that we will check/raise the river if we make our straight. Given our opponentâ€™s holding and the assumptions about his play, this is the best strategy we could employ. In this case, we will now win 6 bets if we make our straight (the 4 in the pot plus the 2 we win on the river), while we still lose only 1 bet if we donâ€™t make our straight. Mimicking the EV calculation from Example 1, the EV of calling is (6 bets) x (0.182) + (-1 bets) x (0.818) = 0.274 bets. Because of the future money won the river when we make our straight, our turn call is now profitable. This is a prototypical example of implied odds.Exercise 1:
Repeat Example 2 if our opponent holds (a) As Kd (b) Ad Qs Â© Ad As. Answers to exercises will be given in Section IV.