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possibly bet the hand on the end for the simple reason that your bet has absolutely no positive expectation. Since your four jacks are exposed for the world to see, your opponent will fold every hand he can have except four queens or a straight flush in hearts. With either of those hands, he will raise. So your bet has nothing to gain and everything to lose.

This very obvious situation points toward the key distinction between play in the final round of betting and in earlier rounds. With one card to come, you would most certainly bet the four jacks to avoid giving your opponent a free card to outdraw you. Your bet forces him either to fold and thus give up any chance to outdraw you or to call and pay for that slim chance. However, when all the cards are out, betting to avoid giving a free card no longer applies. So if you now still decide to bet your hand, you no longer ask what your chances are of having the best hand but rather what the chances are of winning the last bet when you are called

This distinction may seem like hair-splitting, but it is most assuredly not. In fact, it is crucial to successful play — that is, to winning or saving extra bets — when you are heads-up on the end. To take a very common situation, let's say you have three-of-a-kind in seven-card stud, and you know your opponent is drawing to a flush and has nothing else. The odds against that opponent's making the flush on the last card are, we'll assume, 4-to-1, which means you are an 80 percent favorite to have the best hand. However, if your opponent checks, you certainly should not bet because, as in the case of the four open jacks, a bet has no positive expectation. Your opponent will fold if he didn't make the flush, and he will call or possibly raise if he did. So even though you are an 80 percent favorite to have the best hand, you become an underdog if you bet and get called. To repeat, then, the decision to bet a legitimate hand for value on the end should be based not on your chances of having the best hand but on your chances of winning the last bet when you are called.

When you bet for value on the end after your opponent has checked, you must figure your hand has better than a 50-50

chance of winning when you are called. In fact, you have to figure it has at least about a 55 percent chance of winning to compensate for those times when your opponent is planning to check-raise. With three-of-a-kind against a flush draw, you are certainly the favorite, but you are not the favorite if your opponent calls. Yet to show a profit on your last round bets, clearly you must be the favorite even when your opponent calls.

At the same time, you should not carry this principle to such an extreme that you bet only when you have a lock, because then you will not win a lot of final bets you should win. To bet on the end after your opponent has checked, it is only necessary that you are the favorite when your opponent calls. Thus, if you figure you are only a 60 percent favorite when called, you should certainly bet even though you know there's a 40 percent chance your opponent will beat you if he calls. Your bet still has positive expectation. After ten such bets you will have won six and lost four on average for a net profit of two bets. Even if one of those four losses is a check-raise which you call, you still win six bets while losing five for a one-bet profit.

To give a concrete example of such relatively close decisions, let's say you are playing draw poker, and your opponent stands pat and then checks to you when you draw one. Since your opponent stood pat, you are quite sure you are facing a straight, a flush, or a full house. Yet your opponent checked to you. You know he will call with just about any of his hands. Therefore, you should bet an ace-high straight or even a queen-high straight, because your opponent probably would have come out betting himself with a tiny flush or better. Chances are, then, he has a straight smaller than yours. It's true you may lose in the showdown, but you are enough of a favorite with a queen-high straight to warrant a bet.

Last Position Play After Your Opponent Has Bet

Let us now consider your options in last position when your opponent does not give you a free call but comes out betting. When he bets, you can either fold, call, or raise.

Deciding whether to fold or call is relatively straightforward. The question is: Are your chances of winning the pot better than the odds you are getting from the pot, either because your hand is better than your opponent's or because your opponent is bluffing? If you think your chances are better, you call. If not, you fold.

If you are thinking of raising after your opponent bets, you must ask the same question you would have asked before betting had your opponent checked: What are the chances of winning that extra bet when you are called? You should not raise unless you figure you are at least a 55 percent favorite, since you also face the possibility of a reraise. In fact, one way of looking at raising an opponent on the end without the nuts is that you are laying almost 2-to-l odds on that last bet, especially if your opponent is capable of bluffing on a reraise. When you raise and your opponent raises back, you usually lose two bets, but if he calls, you only gain one bet. Of course, this consideration does not apply against a player who will never bluff on a reraise. If such a player raises you back, you can just throw your hand away, knowing you are beat.

Before raising on the end, you must also consider the overall ability of your opponent. Once he puts in an initial bet, an average player will call your raise almost every time. Therefore, you certainly should not try a bluff raise. However, you should raise with any hand you consider a reasonable favorite to win the last bet because you can be pretty sure of getting paid off. Tough players, on the other hand, will frequently come out betting, but they are capable of folding and not paying you off if you raise. Therefore, a bluff raise has some chance against them. However, when you are raising for value against tough players, you should have a better hand than you need against average players, because when the former are willing to call your raise and thus pay you off, they are likely to show down a strong hand. On close decisions you should not raise tough players on the end as often as you would weak or average players because you don't win that extra bet often enough to make the play profitable. Tough players either throw away a hand you would beat or call with a hand you might not be able to beat.

Ironically, though, a raise may sometimes be correct against a world-class player when you have a hand that is only fairly good. The key factor is whether a raise will make your opponent throw away some hands that are better than yours. Let's say you have a hand that you figure has a 52 percent chance of winning if you call, but little chance of winning if you raise and get called. Nevertheless, it would be correct to raise if you think your opponent will then throw away some hands that beat you. If your analysis is correct, a raise might lift you from a 52 percent favorite to a 65-70 percent favorite, and if the pot is big enough, that added 13-18 percent gives the raise positive expectation. Remember, though, that this play is worth considering only against superstars. Against average and good players — and also against superstars most of the time — the basic formula for raising on the end remains the same: Raise only if you are favored to win that extra bet when your opponent calls.

To summarize play in last position after your opponent has bet, you have three options — fold, call, or raise. You should generally fold when the chances of winning are less than the pot odds you are getting. Thus, if your hand has only a 15 percent chance of winning and the pot is $80, you cannot call a $20 bet. However, your chances of winning do not have to be over 50 Percent to justify a call. All that's necessary is that the pot odds you're getting are better than your chances of winning in the showdown. Thus, if you think you have a 30 percent chance and the pot is $80,you would be right to call a $20 bet because the pot dds you're getting are greater than the odds against your showing down the best hand. Even when you decide you can or cannot call with your underdog hand, you have not necessarily eliminated the option of raising. Against a very, very good player, you might consider raising with some mediocre hands if a raise has greater expectation than a fold or a call — that is, if it will make your opponent throw away enough hands that would be better than yours. Anytime you are last and your opponent bets, you always have the three alternatives of folding, calling, or raising. The one that becomes right is the one that gives you the highest mathematical expectation.

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