The Mathematics of Poker

Poker plays can also be analyzed in terms of expectation. You may think that a particular play is profitable, but sometimes it may not be the best play because an alternative play is more profitable. Let’s say you have a full house in five-card draw. A player ahead of you bets. You know that if you raise, that player will call. So raising appears to be the best play. However, when you raise, the two players behind you will surely fold. On the other hand, if you call the first bettor, you feel fairly confident that the two players behind you will also call. By raising, you gain one unit, but by only calling you gain two. Therefore, calling has the higher positive expectation and is the better play.

Here is a similar but slightly more complicated situation. On the last card in a seven-card stud hand, you make a flush. The player ahead of you, whom you read to have two pair, bets, and there is a player behind you still in the hand, whom you know you have beat. If you raise, the player behind you will fold. Furthermore, the initial bettor will probably also fold if he in fact does have only two pair; but if he made a full house, he will re-raise. In this instance, then, raising not only gives you no positive expectation, but it’s actually a play with negative expectation. For if the initial bettor has a full house and re-raise, the play costs you two units if you call his re-raise and one unit if you fold.

Taking this example a step further: If you do not make the flush on the last card and the player ahead of you bets, you might raise against certain opponents! Following the logic of the situation when you did make the flush, the player behind you will fold, and if the initial bettor has only two pair, he too may fold. Whether the play has positive expectation (or less negative expectation than folding) depends upon the odds you are getting for your money – that is, the size of the pot – and your estimate of the chances that the initial bettor does not have a full house and will throw away two pair. Making the latter estimate requires, of course, the ability to read hands and to read players, which I discuss in later pages. At this level, expectation becomes much more complicated than it was when you were just flipping a coin.

Mathematical expectation can also show that one poker play is less unprofitable than another. If, for instance, you think you will average losing 75 cents, including the ante, by playing a hand, you should play on because that is better than folding if the ante is a dollar.

Another important reason to understand expectation is that it gives you a sense of equanimity toward winning or losing a bet: When you make a good bet or a good fold, you will know that you have earned or saved a specific amount which a lesser player would not have earned or saved. It is much harder to make that fold if you are upset because your hand was outdrawn. However, the money you save by folding instead of calling ads to your winnings for the night or for the month. I actually derive pleasure from making a good fold even though I have lost the pot. go play at Online Casino kiwi and try for yourself

Just remember that if the hands were reversed, your opponent would call you, and as we shall see when we discuss the Fundamental Theorem of Poker in the next page, this is one of your edges. You should be happy when it occurs. You should even derive satisfaction from a losing session when you know that other players would have lost much more with your cards.

Pot odds are the odds the pot is giving you for calling a bet. If there is $50 in the pot and the final bet was $10, you are getting 5-to-1 odds for your call. It is essential to know pot odds to figure out expectation. In the example just given, if you figure your chances of winning are better than 5-to-1, then it is correct to call. If you think your chances are worse than 5-to-1, you should fold.

Calling on the Basis of Pot Odds When All the Cards are Out

When all the cards are out, you must decide whether your hand is worth a call, and that depends upon the odds you are getting from the pot and what you think of your chances of having the best hand. It is a judgment problem more than a math problem because there is no way to calculate your chances of winning precisely. If you can beat only a bluff, you have to evaluate the chances that your opponent is bluffing. When you have a decent hand, you must evaluate the chances that your opponent is betting a worse hand than yours. Making these evaluations is often not easy, especially when you have a marginal hand like two pair in seven-card stud. Your ability to do so depends upon your experience, especially your ability to read hands and players. Some things can be learned only through trials by fire at the poker table.

Calling on the Basis of Pot Odds with More Cards to Come

What about deciding whether to call before the draw in draw poker and in stud games when there is one card to come? Now the math becomes important. If you know you have to improve your hand to win, you have to determine your chances of improving in comparison to your pot odds. With a flush draw or an open-ended straight draw-we’ll assume the games is five-card draw poker you would be correct to call a $10 bet when the pot is $50 since your chance of making the flush or the straight is better than 5-to1. Specifically, the odds of making the flush are 4.22-to-1 against and the odds of making the straight, 4.88-to-1 against.

Figuring the odds for making a hand is done on the basis of the number of unseen cards and the number among them that will make the hand. In five-card draw there are 47 unseen cards- the 52 in the deck minus the five cards in your hand. If you are holding four of a suit, nine of the 47 unseen cards will give you a flush and 38 won’t. Thus, the odds against making the flush are 38-to-9, which reduces to 4.22-to-1. If you are holding, say any 6, 7, jack, or queen makes the straight, reducing the odds to exactly 2-to-1 against. Sixteen cards make the hand, and 32 don’t. The smaller the pot odds vis-a-vis the chances of making your hand, the more reason you have to fold. With only $30 in the pot instead of $50, calling a $10 bet for a flush draw or a straight draw (assuming you do not have a joker in your hand) becomes incorrect- that is, it becomes a wager with negative expectation – unless the implied odds are very large, as they might be in a no-limit or pot-limit games.

It is because of the pot odds that people say you need at least three other players in the pot to make it worth paying to draw to a flush in draw poker. With the antes in there, the pot odds are about 4-to-1, and when the bug is used, your chances of making the flush are 3.8-to-1. Notice, incidentally the effect of the antes, the higher they are, the better the pot odds, and the easier it is to call with a flush draw. On the other hand, with no ante and three other players in the pot, you’d be getting only 3-to-1 if you called a bet before the draw, and so you’d have to fold a four-flush then eight of the 47 unseen cards will make the straight — four 8s and four kings -while 39 of the cards won’t help, which reduces to 4.88-to-l.

When a joker or bug is used, as in public card rooms in California, you have an additional card to use to make flushes and straights, which improves the chances of making the flush to 3.8-to-1 and of making the straight to 4.33-to-1. With a joker in your hand, the chances of making a straight improve dramatically; instead of having eight or nine cards to help your hand, you might have 12 or even 16. For example, if you are holding.