Sometimes both you and your opponent have been drawing to a flush or a straight. You don't make your hand, but there's a good chance your opponent didn't make his either. Because of earlier bets on the come, there may be a fair amount in the pot - say, $100 in a $10-$20 game. Now let's say you are first, and you end up with an AJ high. You think there's a 55 percent chance your opponent made a legitimate hand, and there's a 15 percent chance he has you beat "by mistake" with something like an A,K or an A,Q high. In this spot you should bet because by betting you are likely to make your opponent throw away the A,K and A,Q high, thus improving your chances of winning from 30 percent to 45 percent.

In contrast, when you have a busted hand and you suspect your opponent does, too, you may not want to bluff if you end up making something like a small pair. If you bet, your opponent will call with a legitimate hand, and he will fold without one. But if you check and then call, your opponent may bet his busted hands as well as his legitimate ones. Thus, with your small pair you beat his bluffs, which you could not do if you came out betting yourself. Either way, of course, you lose to his legitimate hands.

It is rarely correct to try to bluff out two or more people when all the cards are out; your chances of success decrease geometrically with each additional player in the pot. Paradoxically you might have a profitable bluffing opportunity against each of two opponents individually, but not against both of them as a group. Suppose, for example, you are heads-up on the end in a $10-$20 game. There is $80 in the pot, and you think you can get away with a bluff one out of three times. Clearly this is an extremely profitable bluffing situation. Once you will win $80, and twice you will lose $20 for a net profit of $40 or an average profit of $13.33 per bet. Now suppose you are in the identical situation except that you are up against two players instead of one. We'll assume each player has put $40 in the pot to expand it to $120, and you think,

as in the former case, that each opponent will fold one time out of three. You are now getting 6-to-1 instead of 4-to-1 from the pot. Nevertheless, an attempt at a bluff is no longer profitable because the probability that both of your opponents will fold is 1/3 X 1/3, which equals 1/9. In other words, eight times out of nine one or the other of your opponents will call on average. So you stand to lose $20 eight times for a total of $160 and to win $120 once. Your net loss is $40 or $4.44 per bet. Thus, opposing each individual player by himself results in a profitable bluffing situation, but if they're both in against you, you have gone from a profitable situation to an unprofitable one.

(It should be pointed out that in most bluffing situations against more than one player the probabilities that each player will fold are not independent. The player in the middle will frequently fold a hand that he would call with if he was last, and sometimes the player who is last will call with a hand he would have folded without hesitation had he been in the middle, expecting the player behind him to call. Nevertheless, the general principle still holds that it is usually more profitable to try to bluff one player out of a pot containing 2X dollars than to bluff two players out of a pot containing 3X dollars.)