expectation at Win Poker

When there is only one round of betting left and only one card to come, comparing your chances of improving to the pot odds you are getting is a relatively straightforward proposition. If your chances of making a hand you know will win are, say, 4-to-1 against and you must call a $20 bet for the chance to win a $120 pot, then clearly your hand is worth a call because you’re getting 6-to-1 pot odds. Those 6-to-1 odds the pot is offering you (excluding bets on the end) are greater than the 4-to-1 odds against your making your hand. However, when there is more than one card to come, you must be very careful in determining your real pot odds. Many players make a classic mistake: They know their chances of improving, let’s say, with three cards to come, and they compare those chances to the pot odds they are getting right now. But such a comparison is completely off the mark since the players are going to have to put more money into the pot in future betting rounds, and they must take that money into account. It’s true that the chances of making a hand improve greatly when there are two or three cards to come, but the odds you are getting from the pot worsen.

Reducing Your Pot Odds With More than One Card to Come
Let’s say you are playing holdem, and after the flop you have a four-flush that you are sure will win if you hit it. There are two cards to come, which improves your odds of making the flush to approximately 1,75-to-l. It is a $10-$20 game with $20 in the pot, and your single opponent has bet $10. You may say, “I’m getting 3-to-1 odds and my chances are 1,75-to-l. So I should call.”
However, the 1,75-to-1 odds of making the flush apply only if you intend to see not just the next card, but the last card as well, and to see the last card you will probably have to call not just $ 10 now but also $20 on the next round of betting. Therefore, when you decide you’re going to see a hand that needs improvement all the way through to the end, you can’t say you are getting, as in this case, 30-to-10 odds. You have to say, “Well, if I miss my hand, I lose $ 10 on this round of betting and $20 on the next round. In all, I lose $30. If I make my hand, I will win the $30 in there now plus $20 on the next round for a total of $50.” All of a sudden, instead of 30-to-10, you’re getting only 50-to-30 odds, which reduces to 1,67-to-l.
These are your effective odds — the real odds you are getting from the pot when you call a bet with more than one card to come. Since you are getting only 1,67-to-l by calling a $10 bet after the flop, and your chances of making the flush are 1,75-to-l, you would have to throw away the hand, because it has turned into a losing play — that is, a play with negative expectations. The only time it would be correct to play the hand in this situation is if you could count on your opponent to call a bet at the end, after your flush card hits. Then your potential $50 win increases to $70, giving you 70-to-30 odds and justifying a call.
It should be clear from this example that when you compute odds on a hand you intend to play to the end, you must think not in terms of the immediate pot odds but in terms of the total amount you might lose versus the total amount you might win. You have to ask, “What do I lose if I miss my hand, and what will I gain if I make it?” The answer to this question tells you your real or effective odds.
Let’s look at an interesting, more complex application of effective odds. Suppose there is $250 in the pot, you have a back-door flush draw in holdem, and an opponent bets $ 10. With a back-door flush you need two in a row of a suit. To make things simple, we’ll assume the chances of catching two consecutive of a particular suit are 1/5 * 1/5. That’s not quite right, but it’s close enough. It means you’ll hit a flush once in 25 tries on average, making you a 24-to-1 underdog. By calling your opponent’s $10 bet, you would appear to be getting 26-to-1. So you might say, “OK, I’m getting 26-to-1, and it’s only 24-to-1 against me. Therefore, I should call to try to make my flush.”
Your calculations are incorrect because they do not take into account your effective odds. One out of 25 times you will win the $260 in there, plus probably another $40 on the last two rounds of betting. Twenty times you will lose only $10 when your first card does not hit, and you need not call another bet. But the remaining four times you will lose a total of $30 each time when your first card hits, you call your opponent’s $20 bet, and your second card does not hit. Thus, after 25 such hands, you figure to lose $320 ($200 + $120) while winning $300 for a net loss of $20. Your effective odds reveal a call on the flop to be a play with negative expectation and hence incorrect.

I’ve reread my previous messages… Hmmm, I realized that the “hints” I used to state here are mostly like finger-alphabet… Those who could read them know this for sure. And those who haven’t met this in their everyday live can’t understand anything…
So I decided to explain some poker concepts once again. And I hope I’ll be more precise this time.
Anyway I’m ready to hear your comments and make necessary corrections.
The first thing I would like to deal with is mathematical expectation. The term comes from probability theory and indicates the mean value of the random variable in many identical experiments.
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 runner: 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 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 remember from the Fundamental Theorem of Poker, 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.

1/ You hold 6♣ 5♣ in the big blind. An early player calls, the button raises, the small blind calls, and you call. Four players see the flop of 9♠ 6♦ 3♦. You bet out. The early player calls and the button raises. The small blind calls and you call. All four players see the turn of 5♦. The small blind bets. You raise, the early player reraise, and the small blind calls. There is $320 in the pot. What do you do? (The answer is below  in black font. Mark all the space below in order to know the right answer.)   Answer: Call. You are most likely against a flush. There is also a chance you could be against a straight or a set. You have four strong outs to a full house unless an opponent has 99, or possibly 66 or 55. Four outs are 11 to 1 requiring only a $220 pot for calling to be correct. Note that your raise on the turn was questionable, as a flush was a high possibility given that three opponents paid two bets on the flop to see the turn. In the actual hand, the player called and lost to the early player who showed A♦ Q♦.   2/ You hold Q♥ J♥ in early position. An early player calls, the next early player raises, and you call. A middle player, the cutoff, the button, and the big blind all call. Seven players see the flop of 9♣ 8♠ 6♥. The cutoff bets and the button raises. The first early player calls and the preflop raiser folds. You call and four players see the turn card of K♦. The cutoff bets, the button calls, and the early player calls. There is $430 in the pot. What do you do? (The answer is below  in black font. Mark all the space below in order to know the right answer.)  Answer: Call. Calling bets on the turn for a gut-shot straight is rarely correct unless the pot is quite large. In this hand, the pot is large and your call will close the betting so that you don’t risk a raise behind you. Your four outs are to the nuts since the board is not two-suited or paired. Four outs are 11 to 1 against improving requiring a pot of $330 to justify a call. In the actual hand, the player called and the T♥ fell on the river. He bet and one opponent showed T♠ 9♠.   

3/ You hold A♠ 3♠ in middle position. A middle player limps in and you call. The cutoff, button, and small blind all caps. A wild unpredictable player raises from the big blind and everyone calls. Six players see the flop K♠ Q♦ J♠. The big blind bets, you call, the cutoff raises, and you both call. The turn is the K♣. The cutoff bets and the big blind raises. There is $360 in the pot. What do you do? (The answer is below  in black font. Mark all the space below in order to know the right answer.)  Answer: Fold. You are 4 to 1 against hiding the flush and are getting sufficient pot odds of 6 to 1 on the flush if it would win. However, there are several problems with calling in this situation.   Assuming that your opponents don’t already have a full house, couples of your outs are probably counterfeited and should be disregarded since it is likely that at least one of your opponents holds a king. In this case, you will lose to the Q♠ or to another spade if it matches your opponent’s kicker card. This reduces your outs to seven, which is 6 to 1 against improving and is even money with a pot of $360; however, this is your best case. With a pair on the board, you need to discount your outs for the decent chance that you are drawing dead to a full house. Possible hands of your opponents include KK, QQ, JJ, KQ, and KJ. Some players may add outs for the possible straight; however, you would lose to a full house or KT. Even if the straight won, you would probably only split the pot.   In the actual hand, the player called. The flush came on the river. The big blind folded and the cutoff showed K9. The player collected a large pot in this particular case, but his call has a long run negative expectation given the pot size and betting sequences that occurred in the hand.   

4/ You raise in early position with A♣ K♣. A middle player calls and both blinds call. Four players see the flop of 9♠ 5♦ 3♦. The big blind bets, you raise, the big blind reraise, and you call. The turn is the 2♣. The big blind bets. There is $320 in the pot. What do you do? (The answer is below  in black font. Mark all the space below in order to know the right answer.)  Answer: Call. The worst case scenario is that your opponent has a set or two pair. A straight is doubtful based on the betting sequences in the hand. Even in the worst case scenario, you still have four good outs to a gut-shot straight. A gut-shot draw is 11 to 1 requiring a pot of $440; however, you also have additional outs if your opponent is betting a pair. These outs are counterfeited however if your opponent holds A9, A5, A3, K9, K5, K3, a set, or two pair. I would discount the six outs to the ace or king down to three outs; therefore, J would play the hand as if I had seven outs, which is 6 to 1. Odds of 6 to 1 require a $240 pot; therefore, calling is justified. In the actual hand, the 4♦ came on the river giving the player a straight while his opponent showed 5♥ 3♥.    5/ You hold A♣ K♣ on the button. A middle player calls and you raise. The big blind reraise and you both call. Three players see the flop of Q♠ J♥ 8♠. The big blind bets and you call. The turn is the 9♦. The big blind bets. There is $270 in the pot. What do you do? (The answer is below in black font. Mark all the space below in order to know the right answer.)   Answer: Fold. You have 10 outs to improve your hand; however, an ace is counterfeited or already no good if your opponent holds AQ, AJ, AT, AA, QQ, JJ, or TT. All of these hands are possibilities from a reraise in the big blind. In fact, the only reasonable hand that you could expect to beat if an ace comes is KK. If the river is a king, your situation is worse as you could not heat any reasonable hands, and will only split if your opponent holds AK also. You have four strong outs to the gut-shot straight, although there still is the possibility you might split. Four outs are 11 to 1 and require a pot of $440 to be profitable. In the actual hand, the player folded.    

I’ve noticed that in my previous posts I’ve used some terms that are really important to know.
First of all it’s expected value.
The term comes from the probability theory.
You are in a restaurant looking at a menu. You see two entrees that you like equally, but one is cheaper than the other. You decide to order the cheaper one because you will be just as happy with it. You have just made a decision based on the comparison of the expected value of the two entrees.
You are driving on a highway during rush hour. Your lane seems to be going slower than the lane to your left. The first chance you get, you switch over to the left lane so you can get home faster. You have just made a decision based on the comparison of the expected value of the two lanes.
You are playing poker. The pot is very big, but your hand is mediocre. On the last round of betting, you say “ah, what the heck, I’m going to call, the pot is just too big.” You have just made a decision based on the perceived expected value using information about the size of the pot and the strength of your hand.
Expected value is a concept that everybody uses in their daily lives, although they may not realize it. Whenever we have a choice, we use expected value to guide us on our decision. Sometimes the value of the choices are not purely monetary as it could be based on happiness, a term that academics like to call utility. Usually there is no need to use a formula to calculate the expected value of a decision, but there are some cases where the use of calculating expected value will show us something that is counterintuitive or simply show us why a certain idea is correct or incorrect. It can also help us to pinpoint what factors we need to consider when we are playing poker.
Expected value (EV) is a term used to describe the value of an event over the course of all possibilities. It is an easy way to describe situations that can have many different results, and shows the average result over all the probabilities. A simple example involves a basketball player at the free throw line. If the basketball player has made 750 free throws out of 1000 free throw attempts, you could estimate that he has a 75% chance of making a free throw attempt. Then you can say the EV of the number of points that he will score on one free throw attempt is 0.75. He will either make the free throw and score one point or miss the free throw and not score a point, but on average, he is expected to score 0.75 points with one free throw. The concept of EV is used throughout this book to demonstrate the values of certain poker plays and ideas. This section shows how EV can be calculated and demonstrates how it can be used, in preparation for its usage throughout this book.
The way to calculate the EV of an event is to take all possible events and assign a probability and a result to them. The sum of the probabilities will equal 100%, and the sum of each individual result multiplied by its probability will equal the EV. If the EV of the event is a positive number, we can say the event has a positive expectation or positive value. If the EV of the even is a negative number, we can say the event has a negative expectation or negative value.
Here’s an example in Hold’em
You are playing $10-$20 Hold’em and the pot is currently $80 after the Turn card.. You have an open-ended straight draw and you are 100% sure your opponent has a hand that you will not beat unless you make a straight. But if you do hit your straight, you will win the hand. You believe there is a 17% chance that you will make your straight and a 83% chance that you will not. (In the chapters on Outs and Pot Odds, I will go into further detail on how to estimate your chances of winning and losing.)
Your opponent bets $20 and you must decide to call or fold. You only have $20 left in your stack, and if you call, you cannot lose more or win more on the River as you are considered all-in. If you call and win, you will win $100. If you call and lose, you will lose $20. You have to figure out if calling has a positive expectation.

So you expect to make $0.40 by calling, which means it is better to call than fold. Sometimes you will win $100, more often you will lose $20. However on average, you expect to make $0.40. Calculations like this are difficult for most players to do in their heads while at the poker table. In the chapter on Pot Odds, a simpler way to make the determination of calling or folding is shown. It is practical and much easier to implement, and yet it will be consistent with the EV equations. It is still useful to understand and apply the EV equations when studying the game and thinking about certain situations when not at the poker table. That is its purpose in this book, using it to study the game as opposed to using the equations directly at the table. There are simpler ways to make those calculations and not give up any accuracy.
In poker, whether they know it or not, players are always trying to put themselves into situations where they have positive EV. Good players are able to distinguish between situations that have positive EV and negative EV. When they have positive EV, they will decide to get involved in the hand. When they do not have positive EV, they will get out of the hand. Meanwhile, bad players are not able to distinguish between positive and negative EV. Thus they will often get involved in hands that have negative EV. Sometimes they will get out of hands that have positive EV. Every poker player must identify the difference between positive EV situations and negative EV situations. Once the positive EV situations are identified, the goal is identify the best play that will maximize the EV.

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