Math: No Limit Texas Hold 'em math and math strategy
This article will walk you through the math concepts used in No Limit Texas Hold 'em. It is not meant to be a strategy piece, but several mathrelated strategy issues will be highlighted. We will start off easy and increase the difficulty as we work through some examples. It will also introduce beginning players to some of the poker lingo that will constantly come up in conversations as you talk about hands. More experienced players who have had trouble grasping the math might benefit from a full 360 degrees of it.
The math itself is not particularly difficult to understand and making estimates during play will usually suffice. You will need to become familiar with converting fractions into percentages and vice versa, as well as the peculiarities of betting nomenclature. Most poker betting odds are written and stated in the negative: the odds "against" winning, not the odds in favor of winning. Once you know the odds against you, you can then determine if the expected payoff is worth the bet. It will get a little more complicated when we discuss "implied odds." Although it might feel like more art than science at times, poker is about making educated guesses as to the range of hands that your opponent(s) might be holding and then it is up to you to make correct decisions based on this imperfect information. But before you can do that, you need to know the math. So let's get started.
Percentages, fractions, chances, and odds. There are several different ways to talk about the math probabilities when you consider placing a bet. The first is what I will simply call "chances," as in "what are the chances of hitting my straight" or "what is the chance it will rain today?" "Chances" are usually written as fractions or percentages and are stated in the positive sense. A 1/3 chance means you have a 1 in 3 chance to win. For every 3 attempts, you expect to win 1 time. We can easily convert the fraction into a percentage. A 1/3 chance to win is the same as a 33.3% chance to win. Percentages are really just fractions with a denominator of 100. To check, start with 33.3% which is the same as 33.3/100. Divide the top and bottom by 33.3 and you get back to 1/3. Most people are comfortable with converting the common fractions into percentages and vice versa. Everyone knows 1/3 is 33.3%, for instance. In poker, you will need to know a few more. Memorize these estimates:
The notion of "odds" is a third way to discuss the math probabilities in poker. Colloquially, we often interchange "odds" with "chances" (e.g. "what are the odds it will rain today" is common talk). Oddsspeak, when used in the betting and poker world, usually states the case in the negative. It is a statement of the odds against winning. Using the above example, if you have a 1/3 chance to win, you have a 2/3 chance to lose, or a 2/3 chance against winning.
In betting odds parlance, however, we usually write and say it differently. A 1/3 chance to win, or 2/3 chance to lose, is denoted as "2to1 against," or commonly written as 2:1. The "against" is usually implied, but I will use the term throughout this document to drive the point home. Note that in the "odds against" terminology (e.g. 2:1 against), the first number is the number of times you expect to lose (e.g. 2), the second number is the number of times you would expect to win (e.g. 1), and the sum of the two numbers is the number of total trials (e.g. 2+1=3). If you are facing 2:1 odds [against], you have a 1/3 chance of winning, not a 1/2 chance. That confuses some people. If you are being paid off 2:1 on your money, that means you are winning $2 for every $1 you bet. You get your original $1 returned, along with the winnings of $2. A 2:1 against bet being paid off at 2:1 is considered an even money bet, a coin flip. Understanding this paragraph is critical for making poker decisions based on math.
By the end of this article, you will get comfortable with all 3 ways to discuss the math in hold 'em. Flipping back and forth between fractions, percentages, and odds will also help you think about the strategy behind No Limit Texas Hold 'em.
Familiar examples of odds in casino games. A couple casino examples will help us get familiar with the terminology. A roulette wheel has the numbers 136. (If you actually know how roulette is played, bear with me for a moment.) If you place a bet on any one of these numbers, you would expect to win 1 out of 36 attempts, commonly written as 1/36. Note that a 1/36 chance to win means the same thing as a 35/36 chance against winning. In 36 attempts, you would theoretically win 1 time and lose 35 times. We now know that in betting parlance, we call your odds "35to1 against" or just 35:1.
Now, let's see what the casino will pay us if we win our roulette bet. Lo and behold, the payoff is 35:1. If you bet $1 and you hit your number, you would win $35 (and also get your original $1 bet back). If we played this game 36 times, you would expect to lose your $1 bet 35 times and win $35 on 1 time. After 36 attempts you would break even. Of course, the casino is not so kind, as we shall soon see.
As it turns out, while there are indeed numbers 136 on the roulette wheel, there is also the number 0, and in greedy America, an additional 00. That's 38 numbers in all, not 36. You are still getting paid 35:1 when you hit your number but now it is on a 37:1 long shot bet. That's how the casino makes money and that's why there are only two types of roulette players: losers and liars.
For a second casino example, in craps, the player rolls two 6sided dice. The chance that snake eyes (11), or any specific "hardway," will be rolled is 1/6 x 1/6 = 1/36. The chance against you is 35/36. You would expect to win 1 time out of every 36 rolls. We write this as 35:1 against. The payoff, however, is 30:1. The casino will pay you $30 if you win this $1 bet. For every 36 $1 bets you place, you would expect to lose 35 times for $35 and win 1 time for $30, resulting in a net loss of $5. Although it feels great to win 30 times your bet, it is a sucker's bet, worse than the roulette bet. The house will eventually take all your money.
Poker is about not making these sucker's bet. You play poker against other people, not the house. If you make enough sucker's bets, then the other players will take all your money, just like the casino took all your money in roulette and craps. (In actuality, the casino "rakes" the poker pots too and takes a nice little percentage of every pot so they still take your money, but the difference is that as a poker player, you can still win in the long run.) Poker is a game of skill. If you are good, you will learn to get your money into the pot when the odds are in your favor. If you do that enough times, you will be able to weather the occasional bad beat (losing a hand when you are mathematically way ahead). Casino games, on the other hand, are games of chance, where the odds are always stacked against you. You don't have a choice. You can only hope to put a bad beat on the casino every once and a while.
Poker is also about deception. So, although it is important to know the correct mathematical play in poker, you should not use it at all times since you would then be too predictable. Against casual players, applying the correct math will usually make you a winning player. Against better players, who also know the math, knowing the math yourself is an absolute necessity but not at all sufficient. You also need to read people for strength and weakness (e.g. a bluff). At the highest levels, the cards, betting strategies, and math are all given, and poker becomes primarily game theory and reading opponents.
Side bar: the envelope game. This is a game about information, just like poker. As you know, I run a poker supplies shop in Singapore. Despite the fact that I have a near monopoly on poker supplies here and sell at incredibly reasonable prices, lol, my local customers always ask for a discount. So, I offer them the following deal.
I have 3 envelopes and only I know the contents of each one. One of the envelopes has a big discount coupon and the other two have a lesser amount. The goal is to maximize your chances of getting the larger discount coupon.
Here is the game. I ask you to chose one of the three envelopes. Of the two remaining in my hand, we both know that at least one of them is a small discount. The other is either another small discount or the large one. But for certain, at least one of the two envelopes I am holding is a small discount. I proceed to tear up a smalldiscount envelope, from the two that I am holding, and throw it away. You are still holding your envelope and I am holding the remaining one. I then ask you if you would like to swap envelopes with me. You need to understand the math in order to make the correct decision. Here are the range of answers I usually hear. Which camp do you fall into? Do you switch or not?
 It doesn't matter if I swap envelopes with you. We both have one envelope now and the big discount coupon is in one of them. The chances are 50/50, 50%, 1/2, 1 in 2, a coinflip. It doesn't matter.
 I want to switch. When I originally chose my envelope, I had a 1/3 chance of choosing the correct one. Now, you are giving me a 1/2 chance. A 1/2 chance is better than a 1/3 chance, so I will switch.
 I want to keep my envelope. I know that when I chose it, I had only a 1/3 chance of choosing the correct envelope, and now it looks like you are offering me a 1/2 chance, but I feel lucky. I'm sticking with my first choice. I always go with my gut.
 I want to switch. You must have rigged the game in your favor so I will take the envelope you are holding.
 I want to switch. When I chose my envelope, I had a 1/3 chance of winning. If I switch now, I will have a 2/3 chance of winning the larger coupon. I will therefore take your offer and switch.
What would you do?
Let's think it through. In phase two of the game you only have 2 choices (keep or switch), but you can have several reasons behind your decision. It is important to know the correct reason behind your choice. That way, you will be more consistent in making correct decisions of a similar nature in the future.
You will be amused to know that most people will say something along the lines of #3. They won't switch. Not only will they not switch, their reasoning either reveals a pride of sticking with their first decision, or sticking with it out of confusion once they start thinking about it. "Do I have a 1/3 chance or a 1/2 chance...? Hmmn, I might as well just stick with the one I got."
In this game, you are initially asked to choose an envelope with imperfect information. In fact, you really have no information at all to guide you towards picking the correct envelope. So, you do as would be expected, you take a wild guess and take an envelope. Some people, knowing that I know the contents of each envelope, will try to second guess in which order I have placed the envelopes. "Oh, this might be one of those games where everyone thinks the guy will chose the middle one... or, most people are right handed so it is easier to grab the one on the right... so, I will choose the one on my left." At any rate, your first choice is a guess, regardless of your reason for choosing. In fact, you are better off not having a reason. "I'm not superstitious," they saying goes, "because I've heard it is unlucky." So, just chose a randon envelope. That's all you know right now.
Then, I provide you with new information. The new information is not that I will tear up an envelope. As it turns out, that is rather meaningless. The new information is that you now have a choice to switch with me. The key with new information is what you do with it. Just like in poker, you get new information all the time. You need to decide if it means anything. Have your odds changed? If so, how?
You can probably guess by now that the correct answer is #5. By switching envelopes, you will double your chances of getting the discount coupon from 1/3 to 2/3. As with any brain teaser, only some of the people will be fooled. You want to play poker with those people. If they are your friends, then all the better!
Why is the correct answer #5, to switch envelopes?
If you are not 100% convinced, I can give you two hints. First, would you rather be in my seat, holding 2 envelopes, or in your position, holding just one? After you make your initial choice, I have 2/3 of the envelopes in play and you have just 1/3. That sounds like you would rather be in my seat, so you would happily switch at any time.
Alternatively, play the game in your mind, not with 3 envelopes, but with 1,000 envelopes. You pick one and I keep the other 999 envelopes. I then discard 998 of them that I know are losers. Now, do you want to switch with me? It's still true that the winning envelope is either in your hand or in my hand, which again looks like a 50/50 proposition, but it is even more obvious now that the winning envelope most likely started out in my hand and not yours. I was holding 999 of them to your one. You would rather switch into my seat. The fact that I tear up the envelope(s) is really immaterial and actually quite a diversion. If I continued to hold an overflowing box of 999 envelopes and asked if you want to switch, then it would be more obvious.
There is a lot of new information happening at the poker table all the time. Your job is to figure out what is really new information and what is just noise. Then, use the good new information to inform your decisions. Poker is about making correct decisions and getting your opponents to make incorrect ones.

No Limit Hold 'em math: the rule of 2. Ok, let's finally get to the math behind No Limit Texas Hold 'em. There are 52 cards in the deck. Each card therefore represents 1/52 of the deck, or about 2% of the deck. If you can only memorize one thing, memorize that. For the mathchallenged, notice how if you multiple the top and bottom of 1/52 by 2, you get 2/104, which is darn close to 2/100, which is commonly written as 2%. Once you get your 2 hole cards, you now know 2 of the 52 cards.
Once you see the 3 cards on the flop, you have not seen 47 cards (52 minus the 2 in your hand minus the 3 on the flop). 1/47 = 2.1%, which is still close enough to 2% for me. Hence, think of every card in the deck as having a 2% chance of showing up in any one "slot," either in 1 of the 2 hole card slots in your hand or 1 of the 5 slots on the flop, turn, or river. If you have heard of the "rule of 2," this is it. It is nothing magical, but few books explain where it comes from. Every "out" (a card that will help you win) has a 2% chance of occurring in any "slot," most commonly the turn or the river.
It doesn't matter if the cards are in the deck, in the muck, or in someone else's hand. To us, we have not seen 47 cards once the flop has been flopped and our cards are equally likely to be anywhere. This is not exactly true, if we think our opponent has some of our outs, but we will get to that later.
Preflop statistics. Let's do a little mind calisthenics before tackling the heavy math. What are the chances of being dealt pocket aces? There are 4 aces in the deck and you have a 4/52 chance to be dealt an ace in your first "slot" (the first slot of your 2 hole card slots). If it is an ace, then the chance of getting another ace in your second slot is 3/51. To get aces in both slots, you multiply the probabilities: 4/52 x 3/51 = 12/2652, which equals 0.45%. I can't do that math in my head so a faster way to figure it out is to remember that each slot is worth 2%. That is, you have 2 slots comprising your hole cards and any particular card has a 2% chance of showing up in a slot. If both slots need to be filled by the same type of card, you need to multiply the probabilities. The math looks like this: (4 x 2%) x (3 x 2%) equals 0.48%, which, if you know where to put the decimal place after multiplying 8 times 6, is close enough to the exact answer of 0.45%.
Now that you know you have about a 0.48% chance of being dealt pocket aces, what does that mean? How many hands do you need to see before you get your coveted pocket aces? You should get pocket aces about 1 in every 200 hands. Try to figure out the math before continuing. Our 0.48% estimate can be written as 0.48/100. Try to get the numerator, the top number, to be about 1. In this case, we can multiple top and bottom by 2, like this: 0.48/100 x 2/2 = 0.96/200. That is pretty close to 1/200, i.e. 1 in every 200 hands. Here is another way to calculate it: start with 4/52 x 3/51. That's essentially 12/2500 (note that 5 x 5 = 25 and then just add 2 zeros). Divide top and bottom by 12 and you get a little more than 1/200 (here, 25/12 is about 2 and then add 2 zeros) . At any rate, you will get pocket aces every 221 hands (the exact math).
The speed of a typical live no limit hold 'em game is about 25 hands per hour, so 221/25 = 8 hours. That's right, you should get pocket aces only about every 8 hours of play in a live game. If you play online, where they deal at about twice that rate, you can cut it down to every 4 hours. If you play 4 tables at once, you will get your aces every hour.
If you see pocket aces once in every 221 hands, you will see any pocket pair once in every 17 hands. That's about once per hour. What, you don't see a pocket pair every hour? Ah, you must be unlucky. Why is it 1/17? To be dealt any pocket pair, it doesn't matter what your first card is because the chance of getting the first card is 52/52, or 100%. Then, you just need to match it, which will happen with 3 out of the remaining 51 cards, or 3/51 = 1/17 of the time. In "odds against" language, a 3/51 chance to win is that same as a 48/51 chance to lose, which is "48:3 against," or exactly 16:1 against, which is the same thing as 1/17. Yup, that's a pocket pair every 17 hands. Think of all the hands of Texas Hold 'em you have played in your life. Very few people would agree they have been dealt a pocket pair, on average, every 17 hands. Not that all pocket pairs can be played profitably, but everyone thinks they are unluckier than that.
In the table below, I've listed some of the more interesting probabilities that will give you a sense of what to expect while playing Texas hold 'em. These are preflop statistics. They are most relevant when both players are allin preflop, but just thinking about the numbers will let you know just how far ahead you are or how much you need to catch up. This will also give you a general sense of the strength of various starting hands.
A couple cases from the above chart are worth mentioning. Domination of the higher card is very powerful. For instance, if you have AK against AA, you only have an 8% chance of winning, but AK against KK has a 30% chance of winning. That's almost 1/3 of the time. Against an over pair, any two cards are at best about 20% to win (4:1 against). It hurts a little more if the over pair takes away some of your flush or straight outs. For instance JcJh beats Tc9c 82% of the time but 6d5d only 77% of the time. Well, I guess you are pretty much screwed there either way.
You have a fighting chance to win, almost 1/3 (2:1 against), with middle pairs against just one over card (e.g. 88 vs K5). You can call allin pot size bets in this case if there is dead money in the pot (ie money from players who have folded). That would just be an even money bet, so you would only take it if you were short stacked in a tournament, say.
The more classic preflop coin flip you see all the time on TV is a small pocket pair against 2 over cards. The pocket pair can be favored by as much as 55% in the case of 44 vs A9 (all of different suits), or a slight underdog at 47% if the 44 is up against an over suited connector like JTs. Playing small pocket pairs for a coin flip can be misleading, however. If you put your opponent on two over cards, like AK, then it is true you are in a race situation. But if you put your opponent on a range of hands that includes AK (which would also include a chance she has a bigger pocket pair herself), it is likely that your small pocket pair would be way behind. Any bigger pair crushes a smaller pair 80/20. Keep that in mind when you have a small pocket pair and think you are probably facing a coin flip.
This point cannot be emphasized enough. You know what you are holding for certain, but the best you can do with your opponents is to put them on a range of hands. Don't fall into the trap of thinking, "I thought you were on AK, so my 55 would be a coin flip." You must instead think something like this: "I think you would make that play with any big ace like AK, AQ, or AJ, or any pocket pair from 77 to JJ." Once you weight those possible holdings, your pocket 5's are no longer a coin flip against the range of what your opponent might be holding.
Another interesting strategic point that is not reflected in the "headsup" chart is how preflop matchups play out all the way to the river against multiple opponents. For instance, AA vs any random hand is 85% to win, but against 4 random hands, AA will only hold up 55% of the time, which is just better than a coin flip. The same AA vs just 2 opponents with non conflicting suited connectors (e.g. AdAh vs 9s8s vs 5c4c) is only 58% to win. Again, just better than a coin flip. You need to thin the field with your big pocket pairs and play them against only a couple opponents. You do this by betting aggressively. Sometimes a 4x the big blind bet will thin the field, but sometimes it might take a bigger bet. It depends upon your position (early position raises get more respect), your table image, and the types of players in the hand (tight vs loose). At any rate, it is your job to thin the field with the correct size bet, which might be different every time. If the texture of the flop is threatening (ie 2 of a suit or 2 connected cards), then you really need to pounce on the pot. Aces rarely improve and since the average winning hand in a typical game is two pairs, you need to make sure you aren't playing a typical hand! Whatever you do, don't let several players get to the river cheaply.
Odds on the flop. The preflop odds chart above gives you a sense of how different starting hands will play against each other if they go all the way to the river. This usually is only useful when you are facing an allin bet from a single opponent and need to decide if you want to make the call. It is more likely, however, that you are looking at your 2 hole cards and are wondering what your chances are of hitting some sort of favorable flop.
For instance, if you are holding a pocket pair, you are about 7.5:1 against hitting a set (3ofakind) on the flop. That is about a 1 in 8.5 or 12% chance of hitting. (Using the rule of two, you have 2 outs, each worth 2%, and 3 slots on the flop: (2 x 2%) + (2 x 2%) + (2 x 2%) = 12%.) As we will see later, that means you need to make 7.5 times the money you risk by the end of the hand (if you hit your set) to justify entering the pot to see the flop in the first place. With a smaller pocket pair, if you don't hit your set, you are probably out of the hand. "No set, no bet," as they say. The table below will give you a sense of the likelihood (or not) of connecting various starting hands with the flop.
The first thing that should strike you about the above table is that most any starting hand has a long way to go to improve! The good thing is that your opponents will be in the same boat. This usually means that most of the starting hands that you will probably play in No Limit Texas Hold 'em are pocket pairs and suited connectors (e.g. KhQh).
We already discussed the chance of flopping a set (3ofakind) with your pocket pairs. You are 7.5:1 against hitting a set on the flop and the above table shows that if you can stomach more bets all the way to the river, you will get there only 20% of the time (4:1 against). That's why many people try to limp in preflop with small pocket pairs (call the big blind and hope that there are not any raises) and if they don't flop their set, they shut down. Sets are wellhidden so if you hit, you shouldn't have a problem making some money off them.
With a suited connector (e.g. KQs or 76s), you will flop a made flush or a straight a mere 2% of the time. You will flop a flush draw or an openended straight draw about 22% of the time (11% for the flush draw plus 11% for the straight draw). To connect meaningfully to the flop, therefore, you only have a 1 in 5 chance. You are 4:1 against. And that's to get a flush or straight draw, not the actual flush or straight! A suited connector is much better than just 2 suited cards or just 2 connected cards, however, which won't flop their draws 90% of the time. Granted, playing bigger suited connectors (e.g. AKs or KQs) allows you to connect to the flop by also hitting your cards for pairs. This will happen about 29% of the time. Your lower suited connectors will flop a pair at the same rate, of course, but that pair might not be any good.
In total, big suited connectors will flop their draw or a pair about half the time (22% + 29% = 51%). That sounds a little more promising. And if are holding, say, AKs and you flop top pair with the nut flush draw and a straight draw, then that is a great hand and you will probably want to build yourself a nice big big pot.
The problem with hands like JTs, QJs and even KQs, however, is that they are easily dominated, usually by an ace with one of those cards as a kicker. For instance, if you are holding QJs and the flop comes Qxx, not in your suit nor connected to your straight, then you are very vulnerable. If your opponent is holding AQ or KQ, you have almost no chance to win. You have 3 outs to pair your kicker (the other 3 jacks).
It is worth pointing out that any random starting hand won't connect to the flop (meaning at least a pair) about 2/3 of the time. This is important to remember when playing heads up, where you will need to play almost every hand. Just remember the same holds true for your opponents as well!
While this is not meant to be a strategy session, we can draw several conclusions that might be relevant to No Limit Texas Hold 'em strategy by looking at the above 2 preflop tables:
 Big pocket pairs obviously make strong starting hands. They are at least 75% to win (if played headsup to the river) against smaller cards, whether paired, suited, connected, or both. Their value decreases rapidly against multiple opponents, however, so it is important to thin the field preflop by raising. Apply pressure on the flop as well, especially if the flop comes with 2 of the same suit or 2 connected cards. Don't let your opponent draw cheaply. Make them make a mistake if they are calling you at this point. In a typical game, the average winning hand is 2 pair. It is unlikely you will improve your big pocket pair (you have 2 outs for a set or the board can pair giving you 2 pair) so you need to make sure the hand is not played out as an "average" hand!
 Small pocket pairs will flop a set (3ofakind) about 12% of the time (1 in 8.5 or 7.5:1 against). If you can see the flop cheaply, then these type of hands can make you a lot of money. I would rather flop a set of 4's than 4 aces. The set of 4's is well hidden and should get action, whereas 4 aces is too good of a hand to get any action.
 Suited connectors are popular starting hands, but they will flop their flush or straight draw only 22% of the time. You want to play these hands against several opponents so that more money will get into the pot to make the payoff worth the risk. We will discuss "pot odds" below.
 Most all starting hands need to connect with the flop to have a chance at showdown. That's why most people fold preflop and most of the rest fold on the flop. Otherwise, you will need to bet such that a showdown is avoided (ie everyone folds to you at some point). That is often easier said than done, but the only way it is done is by betting and raising. Most solid players fold about 75% of their hands preflop. If you aren't folding, you should be betting or raising. Checking and calling all the time will not make you a winning player. You need a much stronger hand to check and call than to bet or raise. By checking or calling, you are inviting everyone to play against you. You need to have strong cards (or be on a draw). When you bet or raise, you can win the pot two ways: by everyone folding to you or by having the best hand. The term for this is "fold equity." Much of the time you will have nothing and will need to bet against your opponents' hands, not for your own hand. No Limit Hold 'em is a betting game. Of course, if you are playing 75% of your starting hands instead of folding them, no one will respect your raises. You need to strike a balance between being tight/aggressive (play rarely but bet strongly) and loose/aggressive (play a little more often but still bet strongly). Keep people guessing.
 If 9 people see the flop, a couple players will certainly connect to it. If 23 people see the flop, then only 1 person will likely connect. Poker becomes interesting when one person starts with a big pocket pair, another person connects on the flop and a third player, eying the juicy pot hence created, tries to connect on the turn or river.
 Position, where you are sitting in relation to the dealer button and blinds, is very important in No Limit Texas Hold 'em. It might seem that the odds are always stacked against you but getting to act after your opponents puts you in a much more powerful position. For starters, you can play weaker cards in later positions (nearer to the button, which acts last on the flop, turn, and river betting rounds) since you don't have to worry as much about withstanding later raises as you would have to sweat in early position. In fact, many players will fold every starting hand "under the gun" (the person first to act next to the big blind) except big pocket pairs. You will be "out of position" for the entire hand, so you need to have very strong cards. On the "button" or "cutoff seat" (one to the right of the button), however, you can play a wider variety of starting hands and still keep out of trouble once you see how the action gets to you. While this point is not really mathrelated, it does make the point that you can see the math unfold from a better position when nearer to the button.
 Once the flop is out, you will have seen 5/7, or 70%, of the cards you are going to get. The flop is the defining moment of the hand and gets the most betting action. We will cover postflop odds in the next section.
Postflop odds for an open ended straight draw. Most of the important math comes into play after the flop. A single example will highlight several typical math calculations: the openended straight draw.
Let's assume we are holding 98 suited in early position (just after the blinds) in a No Limit Texas Hold 'em game. We try to limp in (meekly call the big blind), but a player after us puts in a raise and a couple people follow with a call. 98 suited is not a very good starting hand, but it plays well in multiway pots and we feel like playing a little tricky so we call the raise as well. There is $100 in the pot (for sake of discussion) and four players see the flop. The flop comes A76 "rainbow" (i.e. with 3 different suits). We have the nut straight draw (i.e. we are drawing to the best possible straight), but pairing our 9 or 8 is unlikely to help. It is very likely that one or more of our opponents has paired his ace since they are all reasonably tight players who play "big cards." Although we should put our opponents on a range of hands, not a specific hand, let's just assume we need to hit our straight to win the hand.
To summarize, we are playing 98 suited in early position, we called the big blind amount, a player raised and 3 players called (including us), 4 players saw a flop of A76 rainbow, and there was $100 in the pot. Memorize the hand since it will be used for the next 28 pages.
We flopped an openended straight draw. Great. We are now under the gun (we must act first). It is probably best just to check. We will unlikely be able to control this pot by betting right here. The next player, the original raiser, bets $75. That is about 3/4 of the pot, which is quite a normal "continuation" bet. The next player folds and the other player calls. There is now 100+75+75 = $250 in the pot. Should we fold or call this $75 bet? That is the question we will consider for the next several pages. Is it a sucker's bet or a bet with good odds? By the way, raising here would be a bad option since we want to draw cheaply and with a lot of players, which will become evident as we work through the example.
To know whether or not any bet is a good one, we need to figure out the "odds against" us making our straight and how much money we will win if we do hit it, called the "payoff" or usually the "pot odds." If the payoff is better than the "odds against" risk, then it would be correct to call. Said another way, when the casino offers us a 30:1 payoff on a 35:1 [against] bet in craps, we shouldn't take it, but if they were to offer us a 38:1 payoff, then we would gladly accept the challenge, assuming we could even get a spot at the table. So, how do we figure out if we should make the call? It is not as straightforward as reading the layout on a craps table, but here is the 4point game plan:
 Count our outs (the number of cards that will help us make a winning hand)
 Calculate the odds against hitting the winning hand
 Calculate the pot odds (the payoff)
 Compare the "odds against" hitting to the payoff odds
 Calculate the "implied" pot odds and compare again
Step 1: count the "outs." To calculate the odds of making, or not making, our straight, we need to count our "outs," which are the number of cards left in the deck that will help us. We can hit our straight with any of the four 10's or four 5's . That's 8 cards in total, or 8 "outs." That was easy. Move onto step 2 or learn a little more about counting your outs in the side bar below.
Side bar: Discounting outs. counting outs can get more complicated. For a moment, let's say our 98 is suited and 2 of the cards on the flop (A76) are of the same suit, preferably including the ace (think about why before I tell you). We still get it heads up and put our opponent on a paired ace (or on a range of hands, all of which will probably beat us unless we hit). We now have 15 outs. There are 9 cards that will complete our flush and 8 cards for the straight, but one of the 10's and one of the 5's will be in our flush suit. In this case, with 15 outs, we really have some great odds. For instance, we are 15 x 2% = 30% (which is almost 1/3, or 2:1 against) to hit on the turn and 60% to hit on the turn + river. The later means we are actually ahead of the big pair of aces even though we haven't made our hand yet. The trick is to get to see both the turn and the river for a low price.
We would rather hit our straight here than the flush because our suited cards are rather low in rank and we could easily be beaten by a higher flush. In fact, we might not be able to count our flush outs as "full" outs. We need to discount them in case someone else is drawing to a larger flush. For a similar reason, the implied odds, which we will get to later, on the flush must be discounted as well. If we hit our flush and try to get paid off, we would probably only be called by a larger flush. We might call that negative implied odds! Also, flushes are not as hidden as straights and opponents often do not pay them off.
To answer the above question, which ace should be suited? If the Ace on the flop is one of our suits, then by definition, another player could not have paired his ace and have the same flush draw as us. All of our 15 outs are good in this scenario. We might choose to play such a big draw aggressively, as if we were protecting top pair ourselves. This is called a semi bluff. We have nothing at the moment, but we have lots of outs. By betting big ourselves, we earn some "fold equity," which means we might win the pot right there when an opponent doesn't call our big semi bluff bet. For instance, an opponent holding AJ might not call this bet, worried that we might have AK or AQ. If we do get called, we are about even money to win with our big draw if it is checked on the turn or we can see the river relatively cheaply. The primary risk of this semi bluff strategy is to get caught doing it by an opponent who pushes allin with a big ace. If we then call that bet, we are only getting even money, and as I said, there are better ways to get our money into the pot than with a coin flip. The point here, is that if we are the bettor we have 2 ways to win (our opponent folds or he calls but we hit our outs). If we are the caller, we can only win by improving our hand.
Notice that with 15 outs, it becomes much harder for a big pair to push us out of the pot, but it becomes even more important for him to do so. Hence, big draws and big pairs often make for big pot poker. And it is usually a coin flip in the end. Good players try to avoid such a scenario. That is one reason why big draws are often played like made hands.

Step 2: calculate the "odds against" hitting our straight. Once we know how many outs we have, we can calculate the odds of hitting our straight, or, more correctly, the odds against us hitting our straight. (By the way, we are going back to the original scenario where we only have a straight draw, 8 outs.) There are three ways to calculate the odds. It is best to learn how to calculate the odds all three ways since flipping back and forth will come in handy. For now, we will run the numbers for the next card, the turn. We will add the river later.
The first way to calculate the odds is to make an estimate using the rule of 2. We have 8 outs times 2% each. That's a 16% chance to hit our straight. The actual number is 17%, but the rule of 2 gets us close enough. The percentage itself doesn't tell us if we should call the $75 into the $250 pot, however. We need to convert the 16% chance to hit our straight into the "odds against" hitting. There are 2 simple ways to do this. Most people know that 33.3% = 1/3, so since 16% is about half of 33.3%, then you can take half of 1/3, which is 1/2 x 1/3 = 1/6. Or you could have memorized the first table that 16% is about 1/6. Anyway, a 1/6 chance to win is the same as 5:1 against. If you recall, 1/6 means we win 1 time out of 6, which means we lose the other 5 times. Bad:good = 5:1. So, on the next card, we are 5:1 against hitting our 8outer straight. We will use this number to compare it with our payoff that we will calculate in step 3.
The second way is to use fractions. The chance to hit is 8/47, which is derived by dividing the 8 outs by the 47 unseen cards (52 minus the 2 in our hand minus the 3 on the flop). The fraction 8/47 reduces to 1/6 if you divide top and bottom by 8. 1/6 = 5:1. That was fast. By the way, notice that 8/47 is about 8/50, which, if you multiple top and bottom by 2, is 16/100, or 16%, the rule of 2.
The third way is to calculate the odds directly. We start by knowing that 8 cards help us out of the 47 unseen cards. That means 47  8 = 39 cards won't help us. Bad:good = 39:8, which is about 5:1 (by the way, the exact answer is 4.88:1).
Which way is easier? I think it is easier to think of the chance of winning by using fractions or percentages. That's how they do it on TV poker as well. Most of us think of our chances of winning in this way. "I have a one in a million chance to win the lottery." Most nongamblers usually think in this positive case. To tell me I'm 999,999:1 against winning the lottery is not common usage. So, I find it easiest to reduce my percentage or fraction to 1/x (i.e. divide the top and bottom by the top number to make it a 1), and then create the odds against notation as (x1)/1 (i.e. the top number is 1 less than the other bottom number). For example, I start with 8/47 and divide top and bottom by 8 to get to 1/6. Then, I convert the 1/6 into "odds against" by subtracting 1 from the 6 to get 5. That's 5:1 against. It's easy to memorize 1/6 is 5:1, 1/3 is 2:1, etc.
Actually there is a 4th way: You can memorize the numbers in the table below. If you think through your outs in real time, however, you might realize that all outs aren't created equally. You might have to discount some, as we saw above. The table below will illustrate that most draws have 615 outs. For draws with less than 6 outs, it is tough to generate enough money in the pot to justify the risk you would be taking. For super draws with more than 15 outs, you can play them either as draws or made hands (ie as a semi bluff). The most typical draws are openended straight draws and flush draws, which have 8 and 9 outs respectively.
Step 3: calculate the pot odds. In step one, we counted our outs, the number of cards that would likely make us the winning hand. In step two, we calculated the odds against us hitting those outs. Now, in step three, we will calculate the pot odds, the money in the pot in relation to the size of the bet we need to call. There is $250 in the pot and we need to call $75. 250:75 = 3:1. Take the size of the pot, including all bets placed before us in the current round of betting, and divide that number by the amount we need to call. You might think that we should add our current bet (i.e. our $75 call) to the pot as well, but that is not the case. We only can win what is already in the pot, not what we will be putting into the pot when we call. All the money we have put into the pot in previous bets, however, is already in the pot so it is counted. It is no longer "our" money once it is in the pot. If we want it back, we need to win it. That was an easy calculation, but it will get a little more complicated in step 5, when we discuss "implied odds."
Step 4: compare the "odds against" to the "pot odds." We learned that the odds against hitting our straight on the next card, the turn, are a little better than 5:1 against. Our payoff, calculated in step 3 is 3:1. Since the 3:1 payoff is worse than the 5:1 odds against hitting, it would be a bad call in this isolated case. To confirm, if we ran 6 simple trials, we would lose our $75 on 5 occasions ($375 total loss), and win $250 one time. That is a net expected loss of $125. That sounds like a sucker's bet, comparatively worse than the snake eyes craps bet.
The story is not quite finished, however. There are a couple reasons why we would still make this $75 call into the $250 pot on the flop.
If we are allin on the flop, we get to see both the turn and river for one price. Let's assume for now that we only have $75 left in our stack. We are being asked to put in our last $75 to win $250. All the rest of the betting between the other players will go into a side pot, unavailable to us. Since we are putting in our last $75, we will get to see 2 cards, the turn and the river, for the one $75 call. The odds against hitting our straight on the next one card are a little better than 5:1, but since we have 2 chances, our odds are essentially twice as good. Instead of doing the exact math, we can use "the rule of 4." If each out is worth about 2% per slot, then it is worth twice that, or 4%, for 2 slots, the turn and the river. It is not exact, but close enough. To calculate, we have "8 outs twice," or 8 x 4% = 32%. That is about 33%, or 1/3, which we know is 2:1 against. You can also think of 32% as 32/100, which is 32 good and 68 bad, 68:32, or about the same 2:1. Now we would be getting paid 3:1 on a 2:1 against shot. That's a good bet to accept and an easy call to make.
Some people get confused when they try to calculate doubling their odds from just the turn to the turn + river. It is easier to work in the "for" or "positive" chances case instead of the "against" or "negative" odds case. For instance, if we are 5:1 against to hit our straight on the turn, then we are not 5:2 (which reduces to 2.5:1) to hit it on the turn + river. 5:1 means we have a 1/6 positive chance to hit it. We can double the 1/6 chance for hitting on the turn to 2/6 chance for hitting on both the turn and the river. Then, we can flip it back into odds terminology: 2/6 = 1/3 = 2:1. If you insist staying in the odds realm, then doubling your chances on a 5:1 against bet would be 4:2, which reduces to 2:1. Note how we doubled the "good" side from 1 to 2 but kept the total trials the same at 6 (i.e. the sum of the good and bad) so that the "bad" side must be reduced by 1 to 4. At any rate, it would now make sense to make the call, whereas before it did not look like a correct call.
Let's summarize what we have learned so far since it will start to get complicated from here on out. An openended straight draw has 8 outs. We are a little better than 5:1 against to hit on the turn and 2:1 against to hit on the turn + river. In the example, we were being asked to call $75 to win $250, which is 3:1 on our money. We might not make the call when facing a 5:1 shot on the turn, but if we could see both the turn and the river for this one bet, then we would be facing a 2:1 against odds of hitting. Since the payoff of 3:1 is greater than the odds against of 2:1, we would make the call.
Of course, we are only certain we can see both the turn and river cards if we are going allin during the postflop betting round. Otherwise, we might be facing another big bet on the turn from one of our opponents and our odds might be upside down again going into the river. We need another tool to handle this. It is called "implied odds," which we will get to in a second, but first I want to show how the bettor might try to control our destiny.
Side bar: bet to control the odds you are offering your opponent. Let's now consider a slightly different scenario. We still have our same openended straight draw with 8 outs and are still likely up against an big pair, but this time a couple things change. The original raiser, wary that too many players have been smooth calling him all night, puts in a huge bet on the flop and finally everyone else folds to us. For information sake, we both have $500 behind (left in our stacks).
When the original raiser makes a large potsize or larger bet on the flop, perhaps to "protect" his big pair, we can see immediately that he is not laying us the correct odds to draw to our straight, especially if all the other players folded before it got to us. If he bet $125 into the $100 pot and the other players folded, we would be looking at calling $125 to win $225, which is less than 2:1 (225:125 = 1.8:1). Let's say our opponent was the original raiser and is holding something like AK or AQ. If his $125 "overbet" doesn't scare us off our hand, he might fire another big bet into the pot on the turn, denying us the correct pot odds to call again. He is not letting us play our straight draw cheaply. That is good betting on his part if he thinks we are on a draw. If you are not on draw and think you will have the winning hand if the draws do not hit, there is no reason to let the hand go to the river unless the price is very steep. A missed draw is never going to pay you off anyway.
Poker is mostly about making correct decisions yourself and getting your opponents to make incorrect decisions. You cannot change the fate of the cards, but you can influence the decisions you and others make before you see them. The money will follow in the long run. Our opponent is making large bets to force us into making a mistake if we call. He wants to make money on his top pair so in that sense he wants worse hands like ours to call his bets, but he also doesn't want to lay too cheap of a price for us to draw profitably. If we are calling these huge bets hoping to suck out on our straight, on the other hand, we are making bad decisions when the correct mathematical payoff is not there.
In fact, our opponent doesn't even need to have an ace in his hand in this situation. He can "represent" the ace by making this sort of bet. If he reads us for 2 small cards on a straight draw, like we are, then his big bet will knock us off our hand. He can have a medium pair, a big suited connector, or absolute rubbish and make this move, as long as his read on us is correct. This is exactly why I call No Limit Texas Hold 'em a betting game. People think bluffing is such a big deal because you are intentionally trying to lie and be deceptive about the strength of your cards. It seems almost dishonorable, yet perversely respected, to bluff someone out of a hand. If you view no limit hold 'em as a betting game, however, you can place bets against your opponent's hand, not for your hand. If your read is correct, it really doesn't matter what your cards are. Cards are helpful as a backup when your reads are not correct. Taken to a higher level then, poker is about reading other people. It is a people game first, then a betting game with a little game theory thrown in, and then finally a card game.
At any rate, I'm digressing. Let's get back to our original discussion of facing a big bet when the odds don't look like they are in our favor. A savvy reader will have already come up with a big problem in the analysis so far.

Step 5: implied odds. In no limit hold 'em, it is sometimes correct to call some of these apparently poorodds bets. The concept is called "implied odds," and it goes like this. We might not be getting the correct odds to draw to our straight for the money that is currently in the pot ($125 for a $225 pot in the side bar case), but if we think of all the money that might go into the pot by the end of the hand, we might effectively be getting fantastic pot odds. After all, we can only win the pot when the hand is finished, not in mid course. The pot odds are "implied" in this case because we need to make some assumptions about how the hand will play out.
In an extreme [lucky] case, let's say we and our lone opponent both have $500 left in our stack. We call his $125 bet on the flop. We hit our straight on the turn and bet, he raises, we push allin, and he calls. Backing up the hand to the flop, we essentially called his original $125 bet into the $225 pot (explicitly worse than 2:1) to win $725 by the end of the hand (implicitly almost 6:1). Our "implied odds" were 725:125, or about 6:1. Since the 6:1 payoff is better than both our 5:1 odds against hitting on the turn and 2:1 odds against hitting on the turn plus river, it is [was] correct to call the $125 flop bet. This is just one of many possible scenarios we need to consider, admittedly the best one, and if you only think about this one scenario, it will get you into a lot of trouble.
Warning about implied odds. Be careful about implied odds. You can often rationalize to yourself that players will pay you off when you hit your hand and that might not be the case often enough to come out ahead. Playing 98 in big pots against only one opponent is usually not profitable, for instance, even when you hit your straight. You need to consider your opponents and determine who will pay and who won't.
In reality, unless our opponent thinks we were bluffing, he wouldn't likely call off his entire stack to our made straight, and therein lies just one problem. How much will he call? How much do we need him to call in order to make our decision correct to call his $125 flop bet in the first place. Would he even bet once the 3straight hit the board? And, if we don't hit on the turn, how much will it cost us to see the river? Then, if we hit on the river, how much will he pay us off? If we don't hit on the turn or river, is there any chance we can bluff him out of the pot? Those are a lot of questions and we can see how complicated computing implied odds can get. Let's work through some of these issues.
Implied odds hand analysis. The odds against us hitting our hand are largely known and fixed. We know an 8outer is about 5:1 and 2:1 against for the turn and the turn + river respectively. We only need to be sure that all 8 outs will actually win the hand for us. Then, we can easily calculate what the payoff odds need to be in order to justify continuing in the hand: simply greater than 5:1 or 2:1. Many players, however, talk about "implied odds" as if their calculation of them is also a given, as in "I called his bet because of the implied odds." That couldn't be further from the truth. Calculating the implied odds requires educated estimation at best and a lot of hopeful guessing at worst. You are making assumptions. You need to make assumptions about the multiple ways a particular hand will play out and then weigh all of those scenarios before crunching them down to a finite number, the implied odds.
It's difficult to study implied odds without talking strategy. Let's work through the original example again with implied odds in mind. The strategy piece will be largely hypothetical and will be narrowed in scope and possibilities to keep the discussion short. It is presented here merely to give you a taste of some of the things you need to be thinking about when calculating implied odds.
Going back to the original situation, we were calling a $75 bet into a $250 pot on a flop of A76. If you recall, when the original bettor bet $75, it was slightly less than the size of the pot at the time. He was also in a relatively early position so he is representing a strong hand here. The rest of the table folded or made a loose call. By the time it got to us, we were looking at some decent pot odds (3:1) since there were already 2 other callers and one could even think that we could now see the turn card on the cheap.
Scenario 1: we hit our straight on the turn. If we hit our straight on the turn, then it will definitely work out for us. We only put $75 at risk and now know we have the winning hand. We're freerolling the rest of the way. We should be able to make a little money on the hand if it involved 34 players at this point. If we miss our straight on the turn, however, we might face a big bet before seeing the river. Then, even if we hit on the river, we still need to get paid off. Let's see how that might turn out.
Scenario 2: we miss on the turn, face a big bet before the river, and hit on the river. This is where it gets really complicated. It is a common scenario for a straight draw against an aggressivelyplayed big pair. If the turn is a blank (i.e. doesn't look like it would help any draw), we think we will face a pot size bet on the turn from our first opponent, assuming he has an ace, or wants to represent one. He would be protecting his hand. Should we call his potsize bet? Since we called the $75 bet on the flop, there is now $325 in the pot. Our opponent bets the size of the pot on the turn, $325. Let's say everyone else folds around to us, and we call the $325 (we are thinking through this hypothetically to see if this would be the right course of action to take so let's make the call and discuss what might happen). There is now $325 + $325 + $325 = $975 in the pot. We hit our straight on the river (yeah). We are now first to act. We need to get paid off. What do we do?
We are asking this question to see if there is a way to get paid off on our straight, and if so, how much we can expect. We need to think about the answers to these questions before we call that $325 bet. Our hypothetically answers will help us with our decision.
We cannot check here since a checkraise won't likely work. Our opponent is already suspicious that we called his potsize bet on the turn out of position (i.e. we must act before him). He might think we hit our straight (he can see the 3straight on the board), or he might think we slow played a set (3ofakind using a pocket pair, which is wellhidden). Regardless, he knows he only has top pair with a good kicker, and that means he can only beat one pair or an Ace with a worse kicker. All of a sudden, his big pair is looking a little weaker, not just because a 3straight hit the board, but because we are still in the hand. He would be happy to get out of this hand with checkcheck on the river or he might call a small bet. We cannot risk checking ourselves with the hope he will bet so that we can checkraise him. If he checks too, we would only win a small pot and our decision to call the $325 will not have been a good decision in hindsight. We need to get paid off.
We must bet, but how much will he call? We should think beforehand about how much we need to bet and get called on the river in order to make our original $75 and $325 calls the correct decisions. We are not just thinking about this scenario on the river after we hit our straight; we need to think about this possibility way back when we made the $75 call on the flop and certainly the $325 call on the turn. That is the hypothetical case we are discussing.
Backing up to our $75 call on the flop, the "decision point," what were our fully diluted implied odds at that point? Of course, this depends on what we think will happen. The scenario under consideration is that we call the $75, a blank hits on the turn, our opponent bets the size of the pot, $325, to protect what we think is his big pair, the rest folded around to us, and we call the $325. We hit our straight on the river and need to decide what to bet. We ended up putting $75 + $325 = $400 at risk into the pot. We do not count our preflop contribution because it is prior to our decision point (money already in the pot is not ours). Nor do we count our bet on the river since we know we have the winning hand at that point and our last bet is therefore not at risk. We subtract all our bets ($75 on the flop + $325 on the turn + $x on the river), however, from the winning pot size to see what we will win (we can't win our own money if it wasn't at risk).
What does "x," our river bet, need to be to produce pot odds that are better than our 2:1 against odds of hitting our straight by the river? The math is: (net winning amount)/(our bets at risk) > 2/1. The net winning amount, assuming we get called on the river, is (250 + 325 + x) and we put $400 at risk. The equation is therefore: (250 + 325 +x)/400 = 2/1. Solving for x, our river bet, simply multiply both sides by 400 and subtract the left side (250 + 325) from the right side (2x400), or 800  575, which makes x = $225. We need to get a $225 bet called on the river to make our previous decisions mathematically correct. Since there would be $1200 in the pot at that point, the big ace would probably call a $225 bet. He is getting 5:1 on his money, which means he only needs to think we would make all these moves without a made straight about 1/6 of the time.
By the way, a $225 bet in this situation would be considered a "value bet," which is a small size bet relative to the size of the pot made by someone who thinks he has the best hand. He doesn't think his opponent will call a big bet, so he makes a small "value bet," hoping for a call.
So, it is likely our opponent will call a $225 river bet from us. $225 is the absolute minimum, however, that we need to have called by him to justify our previous decisions of calling his bets. That is, our decisions were only "even money," or a "coin flip," at a 2:1 payoff on a 2:1 against shot. There is no reason to put our money at risk for an even money shot. Why play poker just to break even? Don't chase your draws all the way to the river if you believe you are only getting coinflip implied odds. You need a better payoff than that. Reread this paragraph.
I believe you need to get paid off closer to 2.5:1 or 3:1 on the 2:1 against gamble. That means our opponent would need to call a $625 bet by us on the river (for a 3:1 payoff). The math is (250 + 325 +x)/400 = 3/1, for an x = 625). He would then be looking at a $1,600 pot (325 on the flop + 650 on the turn + our 625 on the river), for 2.6:1 on his money. In his mind, we would need to be bluffing or betting a smaller ace more than 1/3.6 = 28% of the time for him to call that bet. It would be a very tough call for him to make, especially if he knows we like to play connectors like 98. If he thinks we were fishing, then he will probably lay down his hand when the straight hits the board and we bet. In such a case, we ended up putting $400 at risk to win only $575 ($250 + $325), which would not pay for our 2:1 gamble back on the flop. That is, we paid dearly for our straight but couldn't sell it to any takers at the end. We won the pot, true, but took too much risk relative to the expected payoff. So, it is likely we would need to make a rather small bet on the river in order to get called. It all depends on what we think this particular opponent will call, hopefully something between $225 and $625. The less hope and the more skill you bring to this guess, the better. If we don't think we can get a decent size bet on the river called, we should have not played the hand in the first place. That is an important message of implied odds. Confidence in our implied odds calculation is critical.
It should go without saying, but many people do not believe this: the result of the hand (whether you win or lose) is not as important as making the right decision. You can't control the cards. They will come and go for you just like for everyone else. The cards are a wash. Only better decisionmaking will make you a better player than someone else.
We can conclude from the above that in the scenario where our opponent bets the size of the pot on the turn and we make the call, and then hit our straight on the river, we would need to get paid off about the size of that same bet on the river. You need to think about your particular opponent and how they would react under these circumstances.
The $75 call on the flop is pretty cheap for a large pot that is being contested multiway. I would make this $75 call all the time, but I probably would not make the $325 call heads up on the turn against a good player who I know can get away from a big pair. I would need at least one other player in the pot to get better payoff odds.
Scenario 3: can we bluff if we miss on the turn and river? Finally, what if we called the $75 on the flop and $325 on the turn but miss our straight on both the turn and river? We have nothing, not even a very high card. Can we try to bluff on the final round of betting? It will unlikely be successful. From our opponent's point of view, we limped into the pot preflop (called the big blind from early position) but also merely called his preflop raise, as did 2 other players between him and us. That doesn't look like we have a hand that we want to get heads up (e.g. AK, AQ, or a big pocket pair) or otherwise we would have raised on one of those occasions. He would put us on a range of hands by trying to figure out what kind of hands we would play this way. Small pocket pairs and suited connectors, for instance, play well in multiplayer pots. Or, he might think we are playing a suited ace with a bad kicker. Let's now play the hand out again and watch how our opponent puts us on a range of hands and then narrows the choices as the hand proceeds. We will see that a bluff is unlikely to work.
After the flop of A76 rainbow, he bet out $75 into the $100 pot. He might make this "continuation" or "feeler" bet as a matter of course. Quite often anyone who shows strength twice on a nondescript flop will take down the pot right there. As it turned out, he got two callers on his $75 bet. That's not too bad with a big ace, but he is going to want to get it headsup on the turn. He doesn't want 2 people drawing against him. Even a bad ace might pair its kicker. By us merely calling his $75 bet, he can eliminate some of our holdings. First, there is no flush draw, but we could still have an ace, although now with a better but still questionable kicker (otherwise we would have raised to see where we are at and to get it headsup). If we were playing rubbish like A7 or A6, then we flopped two pair, but he knows we would never play that out of position with multiple opponents, and we certainly would not continue without a flush draw. He might think we have a straight draw with 98, but unlikely a big gapper like 85 or small cards like 54. He might still think we have a pocket pair. If we have 77 or 66, we just hit a set and are slowplaying it. Any other pocket pair would be afraid of the ace on the board and would probably quit before it got worse. So, I think he puts us on a questionable ace (AJ or AT), a 98 straight draw, or a set of 7s or 6s after we call his $75 on the flop. Notice that we are trying to think about what he is thinking about us. That will help us know what kind of play we can put on him later, if any. Otherwise, he will know we are bluffing.
We then check the turn and he bet $325 to put us to the test. When we call this bet, it is unlikely in his mind that we have a questionable AJ or AT ace. He has to think that we are thinking he will put in another big bet if the river is another blank. The action is too big for us to call yet another big bet so a questionable ace would fold to his $325 bet in the first place. In addition, we have never raised throughout the hand to see "where we are at" with such an ace. A good time to have done this would have been to raise his $75 bet on the flop. Had we had an ace, and especially one with a questionable kicker, we would have either folded or raised to test him and/or to get it headsup. So, our play up to this point makes it unlikely we have a questionable ace. Since we merely call the $75 and $325 bets, he probably thinks we have the 98 straight draw or a set that we are continuing to slow play. Again, see how he is putting us on a range of hands and then narrows it down as the action proceeds.
The river is in fact a blank (in this scenario). The question is whether or not we can bluff with a bet or a checkraise. He was probably a little surprised that we had merely flatcalled the whole way and that we called his potsize bet on the turn. If he fears we were slowplaying a set, a checkraise won't work here on the river since he would be happy to check it down. I don't like that strategy.
The only way we can profitably bluff is to make a bet, and we have to do it in a way that makes him think we have a set. There is 325 +325 +325 = 975 in the pot. Any small bet will be called immediately, unless he thinks we are making a "post oak bluff." This is a bluff that is disguised as a "value bet." That is, a player makes a small bet that usually is begging for a call. The post oak bluff mimics this behavior. A very smart player might not call the seemingly small bet from another smart player capable of making this sophisticated bluff. That's very risky on our part unless we know the player as being very tight and nitty. On the other hand, any large bet will most certainly look very suspicious, too much like a bluff. He would be thinking, "That bet is too large... he [we] must not want me to call it.. so therefore I will call it." Again however, similar to the post oak bluff way of thinking, such a big bet might be made to look like a bluff from someone who actually has a set. Again, again, that is also a risky play for someone who actually has a set because if it is not called, the set does not fill any of its potential. Nonetheless, I don't like a small bet or a big bet for the reasons cited above.
It is more likely that a real set (and remember we are trying to act like we have a set in this scenario) would bet about $500  $700. If I had a real set, I would pick a number nearer to $500 to make sure it got called. If I wanted to bluff, I would pick a number in the higher part of this range. It needs to look like a value bet that is just a tad too expensive for him to call. Whether it works or not is completely dependent on how we each read each other in the situation, the relative sizes of our stacks, our table images, etc. I know a few places where I could pull this off, but not many. That's why I don't think a bluff will work here often enough for it to be a profitable play in the long run.
A more complete answer to the question, "Should we call the $75 on the flop and the $325 on the turn," is to weigh the 3 scenarios above, along with several others we didn't discuss, and come to some sort of weighted implied odds calculation. It will be more art than science. You need to consider various ways that a hand is likely to play out, how much you will need to risk to make your hand, and how much you think you will get paid off. You need to understand multiple levels of thinking and be able to read people.
A good place to start then, is to ask the following questions: "Am I playing against an opponent who I think will pay me off if I hit my hand in this situation?" and "If I miss on the turn, how big a bet will I need to call to get to see the river?" If you are playing against "calling stations" who get married to their big pairs but don't protect them, then you should proceed. On the otherhand, if you are playing against a solid player who can read you like a book and will aggressively protect his hand, then don't even bother.
Postflop summary. To wrap up this section, here are some strategic considerations, grounded in math, for postflop play.
 Postflop betting is usually a struggle between a player with a decent made hand (say, top pair with a good kicker or 2 pair) and a player or two who are trying to draw to a better hand (say, a straight or flush). At least that is what it usually looks like. Quite often the flop will miss one, both, or every player involved in the hand and whoever can "represent" the biggest hand will usually take it down with savvy betting.
 You want to play your draws (small pocket pairs and suited connectors) against multiple opponents so that enough money gets into the pot to make it worth your while. You need the right pot odds. Hopefully the bets will be small but numerous until you get there. Once there, you need to get paid off. Of course, if you get the reputation of being a caller until you hit your hand and then a bet, your predictability will discourage others from paying you off. You need to mix it up and, say, raise about 20% of your drawing hands.
 Conversely, you want to play nondrawing hands (big pocket pairs, top pair, 2 pair) against fewer opponents so there is less of a chance someone can draw out on you. Although you want to make money on your atthetimewinning hands, you will want to make big enough bets and raises so your drawing opponents would be making a mistake by calling you. Of course, this all depends on the "texture" of the flop. If you bring AA to a 3rag board (3 unconnected cards of different suits and of low value), then you will need to slow down your betting to let others catch up. Hands that are just slightly better than other hands make the most money. Monsters rarely find customers.
 Draws are easier to play in late position, where you can see how many players are playing the hand, as well as how much betting action you will need to call (at least on that round) before it gets to you. It is often frustrating to play draws from early and middle position once you call a bet only to be raised later on. You might get priced out of your draw. In earlier positions, however, I will often bet at the flop while on a draw to try to slow down my opponents. I'm essentially setting up my own favorable pot odds with a speed bump that either makes my stronger opponents hesitate or tricks them into slow playing behind me. Sometimes they will just call when otherwise they would have raised. When this works, I get to see my draws a little cheaper. In fact, most novice players, especially whiney losers like Joe, with big pocket pairs like it when other players bet into them. They then think they can just slow play their hand and pop in a raise on the river. Even when you hit your draw, they often just can't get away from their big pocket pair and will pay you off.
 Don't rely on "runner runner" to make your hand. Let's say you have 2 relatively lowvalue suited hole cards and only 1 card of your suit hits on the flop. You need both the turn and the river to be in your suit to complete your flush. That's called "runner runner" or "two running cards." To hit on the turn, you have a 9/47 chance and to then hit on the river you would have a 8/46 chance. You need both to happen so multiplying those two probabilities together results in only a slim 72/2162, or 3% chance of making your flush. That's not likely so save your money.
 Unless you are on a draw as we discussed above, you generally need a stronger hand to call a bet than to raise or initiate the betting yourself. Play aggressively, particularly in late position, and put your opponents to the test, not the other way around. Poker is about being aggressive and making good decisions. The more decisions you force your opponents to make, the more likely they are to make mistakes.
Hold 'em Math Summary. Here is hold 'em math in a nutshell. As detailed above, I'll use an openended straight draw looking at calling a bet on the flop as an example.
 Count your outs, which are the number of cards that are left in the deck that will make your hand. An openended straight draw has 8 outs. Subtract outs if you believe your opponents might be holding some of them. Discount some of your outs if you are not certain they will win you the hand. Don't make this step too complicated though.
 Convert your outs to "odds against." After the flop, there are 47 unseen cards so your odds against hitting your straight on the turn are 39:8 (remember Bad:Good), or a little better than 5:1. By the river, you have 2 shots of hitting. Since 5:1 against is the same as 1/6 for, multiple that positive chance by 2 to get 2/6 = 1/3, which is 2:1 against. You can also use the "rule of 2:" 8 x 2% = 16% chance to hit on the turn, or times 2 for a 32% chance to hit by the river. Respectively, 84:16 = 5:1 and 68:32 = 2:1. You are therefore 5:1 against to hit on the next 1 card and 2:1 for both cards.
 Calculate your pot odds. If you are faced with calling a $75 bet for a $250 pot, your pot odds are 250:75, or a little better than 3:1. In practice, particularly on the flop, this number by itself is not very useful since there will be more rounds of betting and you will likely have to risk more money to get to a payoff that is not entirely known at the moment. You need to consider "implied odds."
 Calculate your implied odds. You might face more betting decisions on the turn if you don't hit your hand. And, hopefully you will get paid off if you do hit your hand. You need to account for all that. Add up all the bets that you think you will need to call or make to finish the hand. Count only the bets that would be "at risk." For instance, if you hit the nut straight on the turn, your turn bet is probably not at risk since you likely already have the winning hand. Then, estimate the net size of the pot you could win (what is currently in the pot plus all of your opponents' bets after the decision point). Divide your net win by your atrisk bets to determine your implied pot odds. Think of a couple different scenarios and weight them accordingly. This is more art than science, but very important to do.
 Compare the odds against hitting your hand to your implied pot odds. If the pot odds are better, then it is mathematically correct to call or bet.
 Math is just one of several considerations to make while playing a hand. The easy part about math is you can learn most of it from a book. You can also learn the math "by feel" through years of experience at the table. It's true that experience is the best teacher, but it is also very expensive!
Please email me with any thoughts or comments. I would appreciate your feedback.
Ken Crouse
www.bluegatepoker.com