Mathematical expectation is the amount a bet will average winning or losing. It is an extremely important concept for the gambler because it shows him how to evaluate most gambling problems. Using mathematical expectation is also the best way to analyze most poker plays.
Let's say you are betting a friend $1, even money, on the flip of a coin. Each time it comes up heads, you win; each time it comes up tails, you lose. The odds of its coming up heads are 1-to-l, and you're betting $l-to-$l. 'llierefore, your mathematical expectation is precisely zero since you cannot expect, mathematically, to be either ahead or behind after two flips or after 200 flips.
Your hourly rate is also zero. Hourly rate is the amount of money you expect to win per hour. You might be able to flip a coin 500 times an hour, but since you are getting neither good nor bad odds, you will neither earn nor lose money. From a serious gambler's point of view, this betting proposition is not a bad one. It's just a waste of time.
But let's say some imbecile is willing to bet $2 to your $1 on the flip of the coin. Suddenly you have a positive expectation of 50 cents per bet Why 50 cents? On the average you will win one bet for every bet you lose. You wager your first dollar and lose Si; you wager your second and win $2. You have wagered $1 twice, and you are $1 ahead. Each of these $1 bets has earned 50 ccnts.
If you can manage 500 flips of the coin per hour, your hourly rate is now $250, because on average you will lose one dollar 250
limes and win two dollars 250 times $500 minus S250 equals a S250 net win Notice again that your mathematical expectation, which is the amount you will average winning per bet, is 50 cents. You have won $250 after betting a dollar 500 times: That works out to be 50 cents per bet.
Mathematical expectation has nothing to do with results. The imbecilc might win the first ten coin flips in a row, but getting 2-to-1 odds on an even-money proposition, you still eam 50 cents per $1 bet. It makes no difference whether you win or lose a specific bet or series of bets as long as you have a bankroll to cover your losses easily. If you continue to make these bets, you will win, and in the long run your win will approach specifically the sum of your expectations.
Anytime you make a bet with the best of it, where the odds are in your favor, you have earned something on that bet, whether you actually win or lose the bet. By the same token, when you make a bet with the worst of it. where the odds are not in your favor, you have lost something, whether you actually win or lose the bet.
You have the best of it when you have a positive expectation, and you have a positive expectation when the odds are in your favor. You have the worst of it when you have a negative expectation, and you have a negative expectation when the odds are against you. Serious gamblers bet only when they have the best of it; when they have the worst of it, they pass.
What does it mean to have the odds in your favor? It means winning more on a result than the true odds warrant. The true odds of a coin's coming up heads are 1-to-l, but you're getting 2-to-l for your money. The odds in this instance are in your favor. You have the best of it with a positive expectation of 50 cents per bet.
Here is a slightly more complicated example of mathematical expectation. A person writes down a number from one to five and bets $5 against your $ 1 that you cannot guess the number. Should you take the bet? What is your mathematical expectation?
Four guesses will be wrong, and one will be right, on average Therefore, the odds against your guessing correctly are
4-to-1. Chances are that in a single try you will lose the dollar. 1 lowever, you are getting S5-to-$ 1 on a 4-to-1 proposition. So the odds are in your favor, you have the best of it, and you should take the bet. If you make that bet five times, on average you will lose $ 1 four times and win $5 once. You have earned $1 on five bets for a positive expectation of 20 cents per bet.
A bettor is taking the odds when he stands to win more than he bets, as in the example above. He is laying the odds when he stands to win less than he bets. A bettor may have either a positive or a negative expectation, whether he is taking the odds or laying them. If you lay $50 to win $10 when you are only a 4-to-1 favorite, you have a negative expectation of $2 per bet, since you'll win $10 four times but lose $50 once, on average, for a net loss of $10 after five bets. On the other hand, if you lay $30 to win $ 10 when you're a 4-to-1 favorite, you have a positive expectation of S2, since you'll win $10 four times again but lose only $30 once, for a net profit of $10. Expectation shows that the first bet is a bad one and the second bet is a good one.
Mathematical expectation is at the heart of every gambling situation. When a bookmaker requires football bettors to lay $11 to win $10, he has a positive expectation of 50 cents per $10 bet. When a casino pays even money on the pass line at the craps table, it has a positive expectation of about $1.40 per $100 bet since the game is structured so that the pass line bettor will lose 50.7 percent of the time and win 49.3 percent of the time, on average. Indeed it is this seemingly minuscule positive expectation that provides casinos around the world with all their enormous profits. As Vegas World casino owner Bob Stupak has said, "Having one-thousandth of one percent the worst of it, if he plays long enough, that one-thousandth of one percent will bust the richest man in the world."
In most gambling situations like casino craps and roulette, the odds on any given bet are constant. In others they change, and mathematical expectation can show you how to evaluate a particular situation. In blackjack, for instance, to determine the right play, mathematicians have calculated your expectation playing a hand one way and your expectation playing it another way. Whichever play gives you a higher positive expectation or a lower negative expectation is the right one. For example, when you have a 16 against the dealer's 10, you're a favorite to lose. However, when that 16 is 8,8, your best play is to split the 8s, doubling your bet. By splitting the 8s against the dealer's 10, you still stand to lose more money than you win, but you have a lower negative expectation than if you simply hit every time you had an 8,8 against a 10.
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 reraise. 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 reraises, the play costs you two units if you call his reraise and one unit if you fold.
Taking this example a step further: If you do not make the Hush 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 chapters. 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 adds 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.
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 chapter, 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.
As suggested in the coin-flip example at the opening of this chapter, hourly rate is closely related to expectation, and it is a concept especially important to the professional player. When you go into a poker game, you should try to assess what you think you can earn per hour. For the most part you will have to base your assessment on your judgment and experience, but you can use certain mathematical guidelines. For instance, if you are playing draw lowball and you see three players calling $10 and then drawing two cards, which is a very bad play, you can say to yourself that each time they put in $ 10 they are losing an average of about $2. They are each doing it eight times an hour, which means those three players Figure to lose about $48 an hour. You are one of four other players who are approximately equal, and therefore you four players figure to split up that $48 an hour, which gives you $12 an hour apiece. Your hourly rate in this instance is simply your share of the total hourly loss of the three bad players in the game.
Of course, in most games you can't be that precise. Even in the example just given, other variables would affect your hourly rate. Additionally, when you are playing in a public card room or in some private games where the operator cuts the pot. you need to deduct either the house rake or the hourly seat charge. In Las Vegas card rooms the rake is usually 10 percent of each pot up to a maximum of $4 in the smaller seven-card stud games and 5 percent of each pot to a maximum of $3 in the larger seven-card stud games, in the Texas hold 'em games, and in most other games.
In the long run a poker player's overall win is the sum of his mathematical expectations in individual situations. The more plays you make with a positive expectation, the bigger winner you stand to be. The more plays you make with a negative expectation, the bigger loser you stand to be. Therefore, you should almost always try to make the play that will maximize your positive expectation or minimize your negative expectation in order to maximize your hourly rate.
Once you have decided what your hourly rate is, you should realize that what you are doing is earning. You are no longer gambling in the traditional sense. You should no longer be anxious to have a good day or upset when you have a bad day. If you play regularly, you should simply feel that it is better to be playing poker making $20 an hour, able to come and go as you please, than to be working an eight-hour shift making $ 15 an hour. To think of poker as something glamorous is very bad. You must think that you are just working as a poker player and that you are not particularly anxious about making a big score. If it comes, it comes. Conversely, you won't be so upset if you have a big loss. If one comes, it comes. You are just playing for a certain hourly rate.
If you have estimated your hourly rate correctly, your eventual winnings will approximate your projected hourly rate multiplied by the total hours played. Your edge comes not from holding better cards, but from play in situations where your opponents would play incorrectly if they had your hand and you had theirs. The total amount of money they cost themselves in incorrect play, assuming you play perfectly, minus the rake, is the amount of money you will win. Your opponents' various mistakes per hour will cost them various amounts of money. If the hands were reversed, you wouldn't make these mistakes, and this difference is your hourly rate. That's all there is to it. If they play a hand against you differently from the way you would play it five times an hour, and if it's a $2 mistake on average, that's a SlO-an-hour gain for you.
To assume you play perfectly is, of course, a big assumption. Few if any of us play perfectly all of the time, but that is what we strive for. Furthermore, it is important to realize that there is not one particular correct way to play a poker hand as there is in most bridge hands On the contrary, you must adjust to your opponents and mix up your play, even against the same opponents, as we shall explain in later chapters.
Furthermore, it is sometimes correct to play incorrectly! You may, for example, purposely make an inferior play to gain in a future hand or future round of betting. You also may play less than optimally against weak opponents who have only a limited amount to lose or when you yourself are on a short bankroll. In these cases it is not correct to push small edges. You should not put in the maximum raises as a small favorite. You should fold hands that are marginally worth calling. You have reduced your hourly rate but have ensured yourself a win. Why give weaker players any chance to get lucky and quit big winners or get lucky and bust you if you arc on a short bankroll? You'll still get the money playing less than optimally. It will just take a few more hours.
You should try to assess most poker games in terms of your expected hourly rate by noticing what mistakes your opponents are making and how much these mistakes are costing them. Don't sit in a game with an insufficient hourly rate projection unless you think the game will become better — either because you expect some w eaker players to arrive soon or because some good players in the game have a tendency to start playing badly when they are losing. If these good players jump off winners, you should quit if you can. However, it is sometimes good to continue in a game with a low hourly rate projection for political reasons — you do not want to get a reputation for gambling only when you have much the best of it. Such a reputation can make enemies, cost you money in the long run, and even get you barred from some games.
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