0. Introduction

This is probably just going to be something that many of the longtime forum regs who know this stuff inside and out can skip. It’s meant to serve as a primer for anyone trying to learn LLSNL who has never seen this stuff spelled out before, or as a refresher for anyone who may have memorized some of this stuff but is rusty with the computations. The intention is not to give you stuff to memorize about poker math, but instead to show you where it comes from so you can do this stuff on your own—which, aside from helping your game, I just think is pretty fun in its own right.

I’m going to skip the extreme basics that everyone should know, but I’ll start maybe one step up from there.

1. Combinatorics

I’ll start by stating the most fundamental principle of combinatorics that applies to poker:

When you have to choose two (or more) things, and you know how many choices you have for each, the way to compute the total number of choices is to multiply.

Let’s see how this is put into practice. Suppose we want to find the number of different combinations of AK that you can be dealt. There are 4 aces in the deck, and 4 kings in the deck. So we multiply 4*4 to obtain 16, which is the answer.

Now let’s say that instead of doing that, we want to instead find the number of combinations of 76s. We start the same way: there are 4 sevens in the deck. But now, after choosing the 7, there is only one 6 that matches the suit. So we multiply 4*1 = 4.

Example: Ordering

Here’s an extremely common example that also applies to poker. Let’s say we have 3 objects: suppose they are cards—an ace, a king, and a queen. We’d like to know how many ways there are to put the three cards in different orders.

If you wish, you can write out all the different ways, but the answer is 6, and here’s a good way to prove it. We have 3 choices for the first card. After choosing that card, we have 2 cards left over for the second card. And that forces the choice of the third card—there’s only one left. By the fundamental principle, 3*2*1 = 6.

This idea, of taking the number of objects we have and computing the number of rearrangements by multiplying every integer from that number to 1, is so common that it has its own name and notation. It’s called “factorial”, and it’s denoted by an exclamation point. The first few factorials are:

1! = 1
2! = 2
3! = 6
4! = 24
5! = 120

(I have never seen an example in poker where I’d want to know a factorial beyond 5!.)

Accounting for Multi-Counting

Now here’s a wrinkle in the above method. Suppose we want to compute the number of ways we can be dealt AA. By the principles laid out above, we should be computing as follows: there are 4 choices for the first ace, and 3 choices for the second ace, so there should be 4*3 = 12 combinations. However, this is not correct in this case. The reason is because we have double-counted. We have essentially claimed that AcAd is a different combination than AdAc. Notice that this did not apply in the above examples when the two cards we were choosing had to be different, so there was no overlap.

There’s an easy way to fix this. All we have to do is take our 12 and divide it by the number of possible rearrangements of the same hand. So in this case, since we have 2 cards, we divide by 2!, which is of course 2. So we get a final answer of 4*3/2 = 6.

Another time in poker when multi-counting comes up is when you try to compute the number of possible flops. We’d like to say there are 52 choices for the first card, 51 for the second, and 50 for the third. But we have multi-counted doing it this way, because the order of the cards doesn’t matter. In this case, we have to divide by 3!, so our final answer would be

52*51*50/3! = 52*51*50/6 = 132600/6 = 22100

The above idea is called the combination function—multiplying a certain number of descending integers and then dividing by that number factorial. It’s (sometimes) denoted by a C. So, for example, 4C2 = 6 (the number of combinations of AA) and 52C3 = 22100 (the number of flops).

Card Removal

Here are 2 more examples that account for a common thing in poker—certain cards being impossible because you have them. Let’s say you have AK, and you want to know if your lone opponent has AA. How many combinations can he have? Well, the computational process is the same as before, but now because you hold one ace, the number of choices for your opponent’s aces has fallen. Instead of 4C2, the answer is now 3C2, which is equal to 3 (3*2/2!).

To revisit another earlier example, instead of computing the total number of flops possible in hold ‘em, let’s instead compute the number of flops you can see after you’ve seen your 2 hole cards. Instead of 52C3, the answer is now 50C3, which is

50*49*48/6 = 50*49*8 = 19600

2. Probability

I assume everyone knows the basic principle of discrete probability, which is that to compute the probability of an event happening, you divide the total number of ways the event can happen by the total number of ways that all events could happen. (If you don’t know what “discrete” means, don’t worry; the important thing to know is that all probabilities you’ll ever see in poker are discrete.) So for example, if it’s the turn, you have a flush draw, and you want to know the probability that your draw will hit by the river (assuming you see your opponent’s hand and there are no blockers in it), there are 9 possible cards that make your flush, but 44 cards total that can come (52 minus the 2 in your hand, the 2 in your opponent’s, and the 4 on the board), so the answer is 9/44.

It turns out that almost every probability problem is exactly this easy on a conceptual level, and usually the only difference is the combinatorics. Let’s try an example. Suppose you hold two unpaired cards and you want to know the probability of flopping exactly 2 pair (using your hole cards). The way to do this is to count all the flops where you make 2 pair, then divide by the total number of flops possible.

The number of flops where you make 2 pair is 3*3*44 (3 for your first pair, 3 for your second pair, and 44 for the non-pairing cards in the deck). That works out to 9*44 = 396.

The number of flops possible for your hand, as computed above, is 50C3 = 19600.

That means the probability is 396/19600, which is approximately 2%. Conceptually it’s just as easy as the flush draw computation; only the combinatorics are harder.

Now let’s do a similar problem but where we have to be careful for multi-counting. Let’s compute the probability of flopping a flush draw with 2 suited cards. The number of flops we will have a flush draw on is (11*10*39)/2. It’s 11 for the suited cards not in our hand, 10 for the suited cards not in our hand or already picked for the first card, 39 for all the other cards, but here we divide by 2 because the order of the flopped suited cards doesn’t matter. (So for example, if we don’t divide, we’re counting Kx8x4y and 8xKx4y as different flops, when we don’t want to.) That works out to 2145.

The total number of flops is still 19600, so the probability is 2145/19600 which is just a shade under 11%.

Computing Probability The Indirect Way

There’s a very common trick in probability that a poker player needs to know. The most common application is to compute the probability that you will hit a draw with 2 cards to come, but there may be others as well. The trick is that the probability of your desired event is exactly equal to 1 minus the probability of all other possible events.

So let’s imagine that we have an overpair, have gotten it in against a flopped set, and we want to know the probability that we will hit our set on the turn or river. The key thing that tips you off that you should be using this trick is the “or” part of the phrasing. In probability, “or” often means you should be adding different probabilities, but in this case it’s easier to take the probability of not hitting, and subtract from 1.

Much like combinatorics, we combine the different probabilities of missing by multiplying.* The probability we miss the turn is 43/45, and the probability we miss the river after missing the turn is 42/44. The product of these fractions is about 91.2%; so the probability of making our set is about 8.8%.

(*Why do we multiply here when sometimes we’re supposed to add different probabilities? The key is that when it’s an “and”, we multiply; when it’s an “or”, we add. In this case we want the probability that we miss the turn “and” we miss the river. In fact, the whole point of why this trick makes things easier is the ability to turn an “or” into an “and” so that we can multiply instead of add.)

This trick is a very good way to see for yourself, by doing some off-table computations, why some poker players use tricks like the “Rule of 4” when they’re at the table. Try computing the probabilities of hitting draws with two cards to come and with 1, 2, 3, all the way up to 15 outs, and compare your answers in percent form to what you would get if you just multiplied the number of outs by 4. (I just showed you the 2-out case.)

3. Odds

At its most basic level, odds is a gambling term that is usually meant to tell you your risk-to-reward when you bet your money. Its relation to probability is that you can convert the probability of an event happening into odds form. While a probability is of the form

(good outcomes) / (total outcomes)

odds are in the form

(bad outcomes) : (good outcomes)

If the probability of an event is A/B, then the odds are (B-A):A. (Easy example: a 1/4 probability means odds of 3:1.) And if the odds are M:N, the probability is N/(M+N).

One important warning about using odds is that you should not do computations with odds the same way you would using probabilities. The notation for odds is such that it’s not meant to be combined with other odds. If you want to do that, convert to probabilities first.

There are basically three ways to use odds in poker: you can take odds, you can offer odds, or you can lay odds.

Taking Odds

This is a pretty common one. You’re thinking of calling a bet, either to draw or because you think your hand is good often enough to win the pot. In this case, your pot odds are: the amount of money currently in the pot, to the amount you have to call. So if the pot was $100, someone has bet $50, and one other person has already called, there is a total of $200 now in the pot, and you have to call 50. You are getting 200:50, or 4:1 (odds reduce just like fractions do). Once you know that, you can convert those odds into a probability—in this case 20%*. If we don’t consider implied odds or reverse implied odds, then we would conclude that you can call if and only if the probability of a good outcome, whatever it is in this case, is at least 20%.

(Obviously implied odds and reverse implied odds are going to be massive considerations; but these two concepts go far beyond basic math so I’m not going to cover them here.)

(*I skipped a step here--I converted the odds to a fraction, 1/5, and then I converted that fraction to a percentage, 20/100. Other than noting that a percent just means "over 100", I will assume I don't have to explain this further.)

Offering Odds

This is an important consideration for when we are betting with what we believe is the best hand and we are trying to get called. The math works just like taking odds, but here it’s important to put ourselves in the shoes of a potential caller. If our pot is $100, and we bet $75, we’re offering 175:75, or 7:3. If we bet 150, we are offering 250:150, or 5:3. And so on. Knowing when you are offering odds to your opponent that are too good is an important part of value betting correctly.

Laying Odds

This is one that’s important when you are betting and hoping to take down the pot right now. We say you are laying the amount you’re betting, to the amount in the pot that you hope to win. This term is usually reserved for when you are betting at least as much as what’s currently in the pot.

For example, let’s say that you’re playing 2/5, and there’s a raise to $25 and a call from the SB. Then you decide to 3bet from the BB to $125 as a bluff. Here we would say you are laying 125:50, or 5:2. Notice that this is not the same as the odds you’re offering your opponents—they’re being faced with calling 100 to win 175, or 7:4. In these circumstances, unlike the other times we use odds, the probability your play needs to work is 5/7, not 2/7.

It would also make sense, if you’re making a pot-sized bluff on the river, to say you are laying even money (1:1), which is another way of saying you need to win 50% of the time to break even. It can make sense to say that a half-pot bluff means you’re laying 1:2, but you rarely hear that terminology in such a case.

4. Expectation

Obviously, expectation is the life blood of any winning gambler. The definition of expectation in the discrete case (which includes poker and all forms of gambling) is: the sum of the products of the various probabilities of events that can happen, times the amount you win or lose when those events happen.

So for example, if you’re facing a half-pot shove on the turn and you hold a 9-out draw, you are getting 3:1 on a call and you will hit 9/44 of the time. When you win, you win 3 bets, and when you lose, you lose 1 bet. So your expectation is

(9/44)(3) + (35/44)(-1) = -8/44 = -2/11 (2/11 is a bit over .18)

Since the answer is negative, you should fold.

There is also another equivalent way to compute your EV. Let's take the same example, but this time I'll show the second method. First you compute what the pot would be after your call--in this case let's say 4. Then, you take the chances of winning that pot, 9/44, times the pot size:

(9/44)(4) = 9/11

Finally, you subtract your contribution to the pot that came at your decision point, which was 1:

9/11 - 1 = -2/11

Notice that of course we got the same answer, since these methods are equivalent.

The ability to compute expectations builds on two things. The first is knowing the probabilities of various events, which is why I wrote about that before this section. The probability of any given card coming on the board or any given runout can always be exactly computed mathematically, as long as you take all available knowledge into account.

The second part of expectation is knowing how much you stand to win or lose every time a different event happens. If you are faced with calling an all-in, or are otherwise close to an all-in, you can compute this accurately, or at least you can accurately arrive at a minimum win or maximum loss so you can decide whether to continue in a hand.

Unfortunately, there are some very important aspects of expectation that simply cannot be computed mathematically. It’s important to recognize what they are, because we never want to be rationalizing a decision using “math” when math actually has nothing to say about it.

The first thing that’s important but can’t be determined by math is the probabilities attached to what your opponent’s cards are. Some part of this is determined by math, but an important part isn’t. For example, let’s say you hold AK and you think your opponent’s range is AA-JJ, AK, and AQ. Mathematically, you can compute the different combinations of cards making up that range; you can assign a percentage of the total range to each hand; and you can even compute the probability that you would win an all-in against that range (on this forum we usually call that your “equity” against the range). However, one important question remains: how did you first assign this range? There could be many valid answers to this question, but they’re going to be based on observations of your opponent’s tendencies. They’re not going to be computational in nature. You need to assign a range first, without using math, and only then can you do more in-depth analysis of that range mathematically.

The second thing that’s important but can’t be determined by math is future action. To show what I mean, let’s take a common example. One of your opponents has raised preflop, and you are heads-up. You hold a small pocket pair and are thinking of calling to set mine. The question you are trying to ask yourself is this: what percentage of the stacks is too big for you to call? (This assumes you will always fold when you miss your set, by the way.)

Well, we can do some math to determine certain things about this situation, but as we’ll see, we can’t get all the way to an answer without bringing in some non-mathematical elements. The first computation is that the odds of flopping a set are about 8:1 (if you want the exact probability, the above methods can help you find it). So if your opponent has raised to 12% of the stacks or more, you need to fold. But that does NOT imply that you should always call if he raised less!

To see why, let’s say that X is the amount you will win if you hit your set. What do we know about what X is? Well, we know that the minimum value of X is the amount your opponent has already put in. The maximum value is the size of your opponent’s stack. Other than that, we know pretty much nothing. We do know that X will probably be different every time, and that really we should be looking for the average value of X. Aside from that, we only have guesses, and those guesses can’t be arrived at by math.

Here are some questions that you need non-math thinking to answer:

-What range is my opponent raising with?
-Is my opponent capable of stacking off with one pair?
-When my opponent is c-betting for value, what sizing will he use?
-Is my opponent going to c-bet as a bluff? If so, how often and for how much money?
-Is my opponent going to bluff multiple barrels?
-What range of hands will my opponent stack off with that my set can beat? What range of hands can my opponent show up with that beat my set?

Notice that once we answer these questions, there is more math we can do to help arrive at an average value of X. “What range?” is the same question covered in the first part above. But aside from knowing our opponent’s range, we also need to know what he will do with that range! It is helpful to know not just how often our opponent will flop an overpair, but how likely it is that we will extract a lot of money from him when this does happen. Even two opponents who will show up with AA the same percentage of the time could lose different amounts of money when they are outflopped.

There are lots of threads on this forum where you can find this question discussed, so I’ll stop here. The point of this was just to remind you that very often your expectation calculations must be based on input that you didn’t get from doing math. I’ll talk more about expectations attached to range analysis in the next section.

5. Conditional Probability/Expectation

Poker is a game with many different decision points in a hand. So if you are going to try to do probability computations in poker, it’s very important that you understand not just probability, but conditional probability. Most people here probably already sometimes do this without thinking about it (in fact we implicitly did some small examples in this thread already), but it’s still worth discussing because I see people get this wrong on the forum sometimes.

To illustrate conditional probability, I’ll use an example from a different game. Craps is a game with a fixed probability of winning. For this example, it’s not important what it is. Let’s say you are playing craps, and your first roll was a 5. From here on out, you will only win if you roll a 5 before you roll a 7. Other rolls do not count.

So if you want to compute your chances of winning after you’ve rolled a 5, that is a completely different calculation than your total chances of winning. To compute it, here is what you do. Since no roll other than a 5 or 7 matters, you can assume that you will either roll a 5 or 7. Now you do a probability computation given that assumption. In this case, there are four ways to roll a 5 and six ways to roll a 7. That means

P(5|5 or 7) = 4/(4+6) = .4

The vertical line in the expression on the left should be read as “given”, i.e. the probability of a 5 given a 5 or a 7.

One common way that this applies in poker is when you assign a range to your opponent, but you hold some of the cards in that range. For example, let’s assume that you raise preflop and your opponent has reraised. You think he’ll do that with AA/KK/QQ/JJ/AK. Now suppose you have TT. Then your opponent’s range breakdown is as follows:

40 total combos
AA 6/40
KK 6/40
QQ 6/40
JJ 6/40
AK 16/40

Those are the probabilities, if you don’t have any blockers, of your opponent having each different hand. Now let’s say you are considering 4betting as a bluff, because you think your opponent will fold everything except AA/KK to your bluff. The probability that your bluff fails is 12/40, or 30%.

But now let’s say you have AK instead. That changes a lot! Now instead of using those probabilities, we have to compute

P(opponent has AA/KK|we hold AK)

Earlier, there were 40 total combos. Now there are fewer. Since we hold an ace and a king, there are 3 AA combos, 3 KK combos, 9 AK combos, and still 6 combos each of QQ/JJ. So this time, there are 6 combos that will call out of a total of 27. That’s a probability of 2/9, just under 22.5%.

Here’s another way that this applies in poker. This one is more subtle. Let’s say that it’s the river, and you are out of position in a $100 pot. Your opponent has $100 behind. You figure that you are behind 50% of the time, and your opponent has air (and will not call a bet with it) the other 50%. Since betting has a negative expectation in this situation, you decide to check. Now your opponent jams all-in. Should you call?

If you reasoned that you should call because you are ahead 50% of the time and therefore calling is +EV, you are incorrect. The reason is because you are computing your expectation against your opponent’s entire range. In other words, you were computing

E(call|50% win, 50% loss)

but what you should be computing is

E(call|opponent bets)

The right answer to the question here is that there is not enough information to decide whether to call. You don’t just need to know how often you are ahead when you check; you need to know how often you are ahead when your opponent decides to bet instead of checking back.

Let’s say that your opponent will always bet hands that beat yours, but will bluff half the time that he has air. Here is how we do this computation. If he bluffs half the time he has air, then that means he bets 75% of the time total: the 50% he’s ahead, plus half of the 50% he’s behind. Now the probability that your opponent has the best hand is

P(opponent is good|opponent bets) = .5/.75 = 2/3

If you extrapolate to compute your expectation, you’ll see that it’s 0.

In that computation we divided by .75; what happened to the other 25% of the total probability? Well, that represents the percent of the time that your opponent checks, and you always win the pot when that happens. For more on this subject, there’s an old COTM on reverse implied odds.

This, or something similar to this, is a mistake I often see people making. They assume that because they were good often enough to continue at an earlier decision point, they must still be good often enough now. The principles of conditional probability dictate that we usually need extra information before we can come to such conclusions. People often say things like “it’s hard to make a hand”, and it’s true—but if someone is only taking a certain action when they have a hand, it doesn’t matter how hard it was to make it, they already have it!

6. Conclusion

This was not meant to be a comprehensive guide to all the math in poker, or a list of things to memorize. It was just meant to give you a sense of how to do the kinds of computations that the strongest players all know how to do. If it seems like this stuff isn’t “advanced” enough, that’s an illusion—as I’ve said on this forum before, all the math that goes into playing good poker is taught in high school. The trick isn’t knowing any super-advanced concepts, but instead knowing when and how to apply the basic stuff.

Thanks for writing this up, CMV. IDK how much discussion it will get, but it's a great reference resource.

Great crash course.

Thx CMV for a nice holiday gift!

If there are other types of basic poker math problems you'd like help figuring out how to solve, please go ahead and post them ITT and OP or other math smart posters will look to add them to the thread.

It surprises me how often regulars don't know these numbers.

venice10 finds it surprising that the avg LLSNL player doesn't know the above....next time you're at a casino that has a royal flush promo, ask if anyone at the table knows the odds of flopping a royal flush if you hold two broadway suited cards. It will surprise you how few players know that there are 19,600 possible flops with 50 cards. They will give you odds of 167,xxx and up & claim they got it off the internet. It's a good way to find out what kind of competition you're up against.

To be fair, knowing the odds of flopping a royal flush, or for that matter most of the stuff V posted (at least, the actual numbers themselves of flopping various things) isn't required knowledge to be a winning poker player. I mean, in a lotta ways you could simply boil down the hit-a-non-junk-speculative-hand poker method to "am I getting in less than ~5% of my stack against idiots who'll likely stack off postflop if I hit" and that should suffice without knowing the actual numbers.

GcluelessmathnoobG

I was speaking about regulars on this forum. Actually on the tables, the situation is much better (for winning money from bad players).

For example,

It is the small leaks in poker that are the hardest to plug. Saying one can call any non-junk hand pf for less than 5% of your stack and be +EV is something I can't even describe in terms of bad play. I'll just leave it at it is effective stacks, not just your stack, that matters.

5% of your stack gives you 20 shots at it. It's 48.x:1 to flop 2 pair with 72o. 2.5% of your stack will give you 50 shots at flopping 2 pair....let's pray variance is on an even keel.

T5o gives us a semi-high card & we can make any straight except one.

The reason I need to know the above percentages, is so that when I fold 96o in the BB, instead of paying 2 BBs to see the flop vs. 5 players & flop top 2 pair, I don't get upset over the fact that I hit my 48.x:1 shot. Or the flop comes 872r etc. Or I fold 97o in the same situation & flop J85r.

I think that one of the 1st huge jumps in my +Ev years ago, was studying Petrov's book on Hold 'Em Odds where he showed you the math. Made it really easy to shrug your shoulders & chuckle inside when your 97o [West Virginia Big Slick], that you mucked preflop, flops a double gutter on a rainbow flop.

Just knowing how often a flop contains 2 of suit, 3 of a suit & all one suit, helps you realize the strength of suited cards. Even if you don't flop a flush draw, that 2nd suit otf may be yours, giving you back-door potential to go with whatever else you have working for you. Heck, just knowing a flop contains 2 of 1 suit more often than it contains 3 different suits should clue you in.

Then there's the odds of flopping a set. So many people quote the 7.5:1 figure. Well if I have 22 and the flop comes 888, I doubt my flopped full house [vs 5 Vs] is worth much of anything, but it's part of that 7.5:1. Tell them they should believe that they can make 15x their pre-flop investment with small/medium PPs & they'll shake their head. They don't believe a set of 5s gets outdrawn that often vs. 4 opponents.

I've seen players take others to the cleaners with garbage like 73o OTB vs. an UTG raise from a Rec on tilt, when the Button has been playing against the guy for hours. However, he isn't playing the odds. Let the flop come A73 like it did & the Rec is shipping his stack when the Button raises. The Rec is going to call the Button's flop raise & when the Turn is an off-suit deuce & the Button hammers again, UTG isn't going to believe him.Because the Button has shown the Rec a bluff or two over the last several hours. The Button 'knew' UTG's range & if the flop came 862r, he knows UTG didn't hit that board, due to his preflop raise sizing & can beat him down post-flop.

However, most of the above has nothing to do with this thread & I don't want to derail it, so I will understand if a Mod chooses to delete it in an effort to prevent an avalanche.

Yeah, I realize that and obviously it is the effective stack (not our stack) that matters and I'm of course oversimplfying the 5% stuff. And I've also realized recently (a little into this year) that I have been playing too loose in my game (finally coming to terms with how much the game has changed over the years and how I've simply overestimated my IO especially OOP). Kinda think I'm on stage III of the 'ol start-out-tight / then-open-up-your-game / then-realize-that-tight-is-right-was-correct-all-along general lifecycle of players (I'm assuming stage IV is busto, lol?).

All I'm saying is that not knowing the exact odds of flopping a royal flush / draw / two pair / etc. isn't necessary to being a winning poker player. It was more a comment towards Zune saying how he can use this knowledge to spot the losing vs competent poker player, which isn't necessarily the case, imo. I've spotted counterexamples I've both, such as math geeks that can regurgitate all these numbers from memory but have a horrible overall method/approach that likely causes them to lose versus clueless math morans that yet have an overall pretty decent approach to poker that likely causes them to win.

G/butdidn'tmeantoderailthetrheadG

I would qualify as a "math geek," yet I don't know most of these numbers venice quoted. I'm also a decent poker player.

IME the people that know these numbers actually tend to be BAD poker players. They try too hard at the wrong things and think knowing these percentages is the key to winning poker. They are focusing on the wrong things.

I believe the greatest use of these stats would be to explain to a losing player why hand selection is important in a pretty tangible way.

^^^^

Yeah, more trying to say what Browni is but he did it better than me.

GcluelessmathnoobG

It isn't about memorizing the numbers to the 4th significant digit. However, you should be aware that your baseline for hitting 2 pair or better with any two cards is about 4%. Making them suited gives you about a 1% chance of hitting a FD and connectors makes it another 1%. You'll get a FD or a SD about 10% of the time.

Keeping those numbers in mind will stop you just calling a raise because, "I could hit a big hand." While you may be able to do OK playing LLSNL without know the math, as you move up, the edge in your game gets smaller. Someone who knows the odds is going to have an edge over someone who doesn't. It will eventually matter. Or you'll end up like all the TV pros who can't compete at high stakes any longer.

When I started learning how to determine odds using Petrov's book, the 1st thing I learned was that there were 19,600 flops with 50 unseen cards. Seems to me that if someone doesn't know that, they haven't conducted a serious study of the odds.

I guess a better example would be people thinking they have a 7.5:1 shot of flopping a set with 22, instead of 8.2:1 & believe that if they can make 9x their preflop investment in a multi-way pot, that they are good to go, because 9 is more than 7.5.

@browni3141: it's not about knowing the exact figure. 48:1 to flop 2 pair is good enough for me. Flopping a flush draw is 8:1. You make a flush by the river 15:1.

2 I really need to know: 1.86:1 to make a flush by the river with a 4 flush otf & 2.18:1 to make a str8 by the river. Must know when considering calling an all-in otf and you believe you have no other equity.

I hardly know the odds of anything, but I know how to calculate the odds of anything. I actually don't even know the odds of completing a flush/straight draw off the top of my head, but I know that 52 total cards minus 5 known cards is 47, with 9 outs to my hand for a flush draw, so my odds are (52-5-9):9 or 38:9, or about 4:1.

I also know that when we have 22 there are 50 unknown cards in the deck. 48C3 flops do not contain a deuce out of a total of 50C3 flops. The odds of flopping a set are (48C3)/(50C3): (1-(48C3)/(50C3)) = (48C3)/((48C3)-(50C3)):1 = 7.5:1 against. How did you get 8.2:1?

7.5:1 includes those times you flop a full house on a flop of 555. How good do you think your 22 is vs. 5 Vs on a flop of 555?

Then there's those times you have 99, flop is TT9 & V has T9, or, he has JT and gets there OTT.

It doesn't include those types of flops. It accounts for all flops that have at least one deuce. It does include flops like 2h3h4h where our hand is very far from the nuts and hands that give us action will likely have decent equity, however. Let's look at the math again.

Like you said the total number of flops when accounting for card removal is 19600. We want to find the number of flops which have at least one deuce. For this we can recognize that it's equivalent to 19600-(flops containing no deuces) and use the "choose" function. If you're not familiar with this I recommend researching it. Explaining it would make this post a little long. There are 48 non-deuce cards from which we are choosing 3 for the flop. There are (48 choose 3) flops containing no deuces. Note that this includes flops like 555, and we are EXCLUDING these flops when we do 19600-(48 choose 3) = 2304 flops containing at least one deuce, and 48 choose 3 = 17296 flops containing no deuces. The odds we flop a set are 2304:17296, or about 7.5:1 against.

This is part of my earlier point. The math is much more complicated than knowing odds. Like you said we're not stacking villain with 100% equity the 1/8.5 times we manage to flop a set. Often he'll have bluffs that fold or get away from the worse hand, have us beaten already or have equity against us. The EV equation of calling a raise/3-bet with a small pocket pair is complicated.

You're right!
Quads = ((2/50)*(1/49)*(48/48))*3 = 0.24%
Full House = ((2/50)*(48/49)*(3/48))*3 = .73%
Set = ((2/50)*(48/49)*(44/48))*3 = 10.78%

(100-11.75)/11.75 = 7.51:1

Again, I'm just addressing the point of being able to identify a good player based on whether he knows these odds, and overall I think it's a pretty useless metric. For example, here there really isn't that much difference between thinking it is 7.5:1 vs 8.2:1; the difference is pretty unimportant, and I couldn't care less if my opponent could quote the correct one to me. The only thing that matters is whether my opponent believes getting ~9:1 is good enough to setmine, which it clearly isn't (and he's making a huge mistake thinking that he's going to get paid off 100% of the time / not get sucked out when ahead / not flop a set and be outflopped).

I understand what V is getting at, especially with regards to edges being thinner as our opponents get better, so I suppose the more you know the better chance at having a thin edge. FWIW, I have zero clue how the inner workings of a car works, although my guess is that I'm a much safer driver than most who do.

G/derail,imoG

Good OP ruined by a few idiots posting useless nonsense again. Every post after OP needs to be deleted.

still reading thru, but the laying odds section appears to have an error, 2/5 raise to $25, called by SB, and you 3! from BB, you are laying 120:55 because $5 is already in the pot from your blind.

Oh, yeah, you're right. I just wanted to give an example with easy numbers, so I made it so the blinds aren't extra dead money. But yeah, you put in $120 to win $55.

Thanks for writing this up, very good explainations!

I know maths are important and I am not arguing that, still most of this is probably useles and is definetely not the missing puzzle for the old pros to be competitive at the high stakes tables today. ( ivey and dwan probably are playing the highest stakes and the rest we do not know, I would assume they are still great away from the public eye)

What is important is to be able to calculate while at the tables without needing to take out your calculator and work out what 9*9(8/47) - 3+3(4) means.

What is most important and we need to know as LLSNL players is :

-pot odds, and use the most simple way there is to work it out.
-impplied odds, and use the most simple way there is to work it out.
-counting outs to work out if the odds you are being offered are good.
-counting outs to work out what odds you should offer.
-few more concepts but on a lower scale of importance.

The rest of the numbers OP mentions is just fun knowledge. Ofcourse if we were to play the highest stakes out there, we would definetely need to spend more time and study statistics and maths related to poker in depth, still the content of the OP would not account for much of our study. Poker is not Algebra.