Analyzing the Cost of a Mistake

Unfortunately, the play that is likely to be right most of the time is not always the correct play. When you have a choice of plays, you also have to decide how bad it will be if you make a mistake. Here is an obvious example. If your opponent bets on the end and you think the chances are better than 50-50 that that opponent has the best hand, the correct play most of the time is to fold and save a bet. However, it costs you not just one bet but the whole pot when folding turns out to be a mistake - that is, when you fold the best hand. Therefore, you would call, even though the chances are that you are making a mistake. The reason you call is that this mistake costs you only one bet, while the opposite mistake - folding when you have the best hand - costs you the whole pot. (This is simply another way of stating that you should call when the pot odds you are getting in relation to your chances of having the best hand make calling a play with positive expectation.)

There are other situations, as well, where making the wrong play can cost you a considerable amount of money, so you should not necessarily choose that play though it is favored to be right over 50 percent of the time. Such situations come up particularly in no-limit poker. Suppose, for example, you have two queens in no-limit hold 'em, and you put in a small raise before the flop. Everyone folds except one player, who fires back with a gigantic reraise. You know that this player will make such a play not only with two aces and two kings but also with ace, king. Assuming you have nothing other than Bayes' Theorem available to put your opponent on one of these three hands, the odds work out to be 4to-3 in favor of your opponent's having ace, king rather than a pair of aces or a pair of kings. Thus, 4/7 of the time your pair of queens is the favorite, and 3/7 of the time it is the underdog. However, when your opponent does have ace, king, your queens are only a 13-to-10 favorite since there are five cards to come, any one of which could give your opponent either a pair of kings or a pair of aces. So while you will average winning 13 times, the other 10 out of 23 times you will lose the hand when you call the raise and your opponent has ace, king. On the other hand, those three times out of seven when your opponent has two aces or two kings, your two queens are a big 41/2-to-1 underdog, meaning in those instances you will lose 18 hands out of every 22 you play on average.

Therefore, you cannot say, "My queens are 4-to-3 favorites to be the best hand. So I must call." It works out that the 3/7 of the time your opponent has two aces or two kings, you hurt yourself so much that you don't gain it back the 4/7 of the time when he has ace, king.

The general principle operating here is the following: When one alternative will have slightly bad consequences if it's wrong and another second alternative will have terrible consequences if it's wrong, you may be right to choose the first alternative even when the second is slightly favored to be the correct play.

Your opponent bets $30, and you know this opponent will bet anything in this spot except two pair. Should you call or raise? Probability tells us your opponent is a slight favorite - about 55 percent-to have his 8,7 low made when he bets, assuming he started with three small cards. When he does have an 8,7 low, you should not raise since you are a slight underdog and will probably get reraised. However, when one of your opponent's up cards has paired one of his hole cards the remaining 45 percent of the time, a raise is very profitable since you are a big favorite. Thus, a call is correct 55 percent of the time, and a raise is the better play 45 percent of the time. Nevertheless, the best play is to raise because raising will be slightly wrong 55 percent of the time, but calling will be very wrong 45 percent of the time. In other words, even when your opponent does have an 8,7 made and reraises, you still have a good chance of outdrawing him. However, when he has paired, he has only a slim chance of beating you since your 91ow is already the best hand and you have an excellent chance of improving to beat your opponent - even if he makes his 8,7. In the long run then, you do better by raising than by calling though raising will be right only 45 percent of the time.