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 12-02-2017, 02:11 PM #2 Garick Oberbiergenießer     Join Date: Dec 2007 Location: Do you even math, bruh? Posts: 17,683 Re: COTM: Basic Poker Math Thanks for writing this up, CMV. IDK how much discussion it will get, but it's a great reference resource.
 12-02-2017, 02:16 PM #3 QuantumSurfer old hand     Join Date: May 2012 Location: Home on the Range Posts: 1,628 Re: COTM: Basic Poker Math Great crash course.
 12-04-2017, 04:54 PM #4 samo veteran     Join Date: Aug 2010 Location: NJ Posts: 2,902 Re: COTM: Basic Poker Math Thx CMV for a nice holiday gift!
 12-05-2017, 12:29 PM #5 Garick Oberbiergenießer     Join Date: Dec 2007 Location: Do you even math, bruh? Posts: 17,683 Re: COTM: Basic Poker Math 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.
12-06-2017, 08:06 PM   #6
venice10
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Re: COTM: Basic Poker Math

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

Quote:
 Originally Posted by RapidEvolution I was pretty sure that looking at all the postflop stuff would be good, but I also thought there wasn’t a whole lot of discussion about why we choose the hands we choose preflop, so this post will cover both aspects of the game. The preflop stuff is mostly mathematical with some reads, and the postflop stuff is mostly reads with some math…any surprises there? There shouldn’t be!!  Preflop So…what’s a drawing hand? Basically, it’s a hand that you don’t rate to be the best right now, but has the potential to become something huge later. To be fair, any hand has the potential to become something huge, so let’s take a look at some hand-types and see how likely each of those is! This first set deals with sitting with any two unpaired cards and we’ll consider “huge” to be 2pair plus. If we have two random, unpaired cards: flopping EXACTLY two pair by pairing EACH of your hole cards 2.02% flopping EXACTLY trips by flopping two cards to one hole card 1.347% flopping EXACTLY a full house, trips of 1 hole card and pairing the other 0.092% flopping EXACTLY four of a kind, three cards to one of your hole cards 0.01% flopping four to a flush 2.245% If we have any 2 suited cards, we figure the first 4 plus: flopping a flush (including the slight chance of a straight flush in some cases) 0.842% flopping four to a flush 10.944% If we have connectors: flopping a straight (including the slight chance of a straight flush in some cases) 1.306% flopping an 8 out straight draw* 10.449% If we have one-gappers: flopping a straight (including the slight chance of a straight flush in some cases) 0.980% flopping an 8 out straight draw* 8.08% If we have a pair: flopping EXACTLY two pair by pairing the board 16.163% flopping EXACTLY trips by flopping a set for your pocket pair 10.775% flopping EXACTLY a full house, a set to your hole pair + pairing the board 0.735% flopping EXACTLY a full house, by the board tripping up 0.245% flopping EXACTLY four of a kind, two cards to your hole pair 0.245% Dealing with SC’s is HIGHLY math intensive…fortunately, this amazing thread has done it. http://archives1.twoplustwo.com/show...0&fpart=1&vc=1 This is a lot of info to take in, but the general idea is that while any two cards can flop a huge hand, the likelihood of flopping something big with 27o is much lower than with 55 or 67s. Think about it. With 78s we can hit all the same types of hands that 27o can hit, but also hit straights (and straight draws), flushes (and flush draws) and occasionally straight flushes. Looking at it from an equity standpoint: Text results appended to pokerstove.txt 41,095,296 games 0.016 secs 2,568,456,000 games/sec Board: Dead: equity win tie pots won pots tied Hand 0: 22.475% 22.29% 00.18% 9161616 74394.00 { 76s } Hand 1: 77.525% 77.34% 00.18% 31784892 74394.00 { AA } --- 123,285,888 games 0.016 secs 7,705,368,000 games/sec Board: Dead: equity win tie pots won pots tied Hand 0: 11.800% 11.59% 00.21% 14288040 260154.00 { 72o } Hand 1: 88.200% 87.99% 00.21% 108477540 260154.00 { AA } It shouldn’t be much of a surprise that a suited connector holds up much more often than 27o, but realize that a HUGE percentage of this difference is from the draws (mainly because they’re both 7-high hands, so the high-card value is nearly the same. For those of you who are interested, 56s has 22.5% equity vs AA and in terms of sole high-card value, 72>56). Why does any of this matter? Well, assuming we’re going to play straight-forward (meaning that we’re folding if we totally miss), we’re going to need to make a good deal of money when we do hit (in order to counter the losses we incur when we miss). The amount of money we need to earn can be roughly calculated by: Money made > preflop investment x (100\percent chance of hitting). If you’ve heard of the 5 and 10 rule, or the 3 and 6 rule, this equation is where those rules came from. Let’s look at each aspect of this equation. Money Made So…what determines how much money we can make? a) Stack sizes: Obviously, the maximum we can win is the villain’s (or villains’) stacks. b) Villain range: The stronger the villain’s range, the more likely he is to be willing to put the monies in! c) Villain tendency: Some villains don’t like to fold…some fold a whole lot. Those that fold less will give us more money (assuming we’re playing straight-forward) d) Board texture: If the board looks really scary, our villain may find a reason to fold. If it looks safe, he may be more likely to pay us off. e) Our image: If we look really tight/nitty, an observant player will fold to us more often than if we’re raising and betting all over the place. Preflop Investment: What do I mean by “preflop investment”? Basically, I mean the amount of money you put in preflop. (Whether you’re calling a raise, or limping in). It should make sense that the equation is easier to satisfy as our preflop investment goes down. In fact, some hands that are highly profitable to play for a small investment, are worthless when the “price of poker” gets too high. Note that this is also directly related to the effective stack size. Let’s look at a couple of situations. Pokerstars \$1/\$2 (assume all other players are 100BB deep) Villain (\$ 120) (UTG) Hero (\$120) (BTN) Hero has 6h 6d Villain raises to \$xx Hero…? If we look at our equation, then money made needs to be > investment x (100/10.8) Investment x (9.26) Since the cap on the money made is \$120 (assuming everyone folds), then the most we should be willing to invest is \$12.96. Of course, this assumes that villain is stacking off on every flop (which isn’t the case) so, it should actually be less than that. (The rule of 5/10 says we should be willing to pay \$12 max, given good reads). In essence, if the villain raises to \$13 or more, we should be folding now if our plan is to fold the flop if we miss. However, if the stacks are \$250, we can call a much larger raise and still be able to make monies! Percent Chance of Hitting: We’ve covered that, pretty much! Feel free to take a look back for some percentages, or….. CLIFF NOTES (rougly)! Hitting with jank: 5.714% Hitting a straight or straight draw with connectors: 11.85% Hitting a set with a PP: 10.8% Hitting a flush or flush draw with sooted cards: 11.85% Flopping sexy with a SC (sd, fd, combo draw, or made hand: roughly 25% Note: Flopping sexy is not my term, but it’s awesome.

 12-07-2017, 09:17 AM #7 ZuneIt veteran   Join Date: Mar 2009 Location: Where I can find out how2play poker Posts: 2,155 Re: COTM: Basic Poker Math 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.
12-07-2017, 02:45 PM   #8
gobbledygeek
Poet Laureate of LLSNL

Join Date: Jul 2006
Posts: 27,671
Re: COTM: Basic Poker Math

Quote:
 Originally Posted by ZuneIt 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

12-07-2017, 07:39 PM   #9
venice10
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Re: COTM: Basic Poker Math

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,

Quote:
 Originally Posted by gobbledygeek 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
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.

 12-07-2017, 11:50 PM #10 ZuneIt veteran   Join Date: Mar 2009 Location: Where I can find out how2play poker Posts: 2,155 Re: COTM: Basic Poker Math 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.
12-11-2017, 01:25 PM   #11
gobbledygeek
Poet Laureate of LLSNL

Join Date: Jul 2006
Posts: 27,671
Re: COTM: Basic Poker Math

Quote:
 Originally Posted by venice10 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.
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.

Last edited by gobbledygeek; 12-11-2017 at 01:31 PM.

 12-11-2017, 03:28 PM #12 browni3141 veteran   Join Date: Aug 2015 Location: South Florida Posts: 2,418 Re: COTM: Basic Poker Math 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.
 12-11-2017, 04:49 PM #13 gobbledygeek Poet Laureate of LLSNL   Join Date: Jul 2006 Posts: 27,671 Re: COTM: Basic Poker Math ^^^^ Yeah, more trying to say what Browni is but he did it better than me. GcluelessmathnoobG
 12-11-2017, 07:26 PM #14 venice10 Referee     Join Date: Nov 2007 Location: Nowhere special Posts: 22,668 Re: COTM: Basic Poker Math 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.
 12-11-2017, 10:54 PM #15 ZuneIt veteran   Join Date: Mar 2009 Location: Where I can find out how2play poker Posts: 2,155 Re: COTM: Basic Poker Math 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.
12-12-2017, 02:54 AM   #16
browni3141
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Re: COTM: Basic Poker Math

Quote:
 Originally Posted by ZuneIt 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?

12-12-2017, 04:20 AM   #17
ZuneIt
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Re: COTM: Basic Poker Math

Quote:
 Originally Posted by browni3141 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.

12-12-2017, 04:47 AM   #18
browni3141
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Re: COTM: Basic Poker Math

Quote:
 Originally Posted by ZuneIt 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?
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.

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

 12-12-2017, 05:28 AM #19 ZuneIt veteran   Join Date: Mar 2009 Location: Where I can find out how2play poker Posts: 2,155 Re: COTM: Basic Poker Math 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
12-12-2017, 01:15 PM   #20
gobbledygeek
Poet Laureate of LLSNL

Join Date: Jul 2006
Posts: 27,671
Re: COTM: Basic Poker Math

Quote:
 Originally Posted by ZuneIt 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.
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

 12-14-2017, 03:16 PM #21 wait journeyman   Join Date: Oct 2017 Posts: 399 Re: COTM: Basic Poker Math Good OP ruined by a few idiots posting useless nonsense again. Every post after OP needs to be deleted.
 12-15-2017, 03:51 AM #22 Tomark veteran     Join Date: Mar 2010 Posts: 2,525 Re: COTM: Basic Poker Math 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.
 12-15-2017, 09:45 AM #23 CallMeVernon COTM Crusher   Join Date: Jun 2010 Location: Not Vancouver, BC Posts: 3,130 Re: COTM: Basic Poker Math 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.
 12-18-2017, 04:17 PM #24 LongDTravis journeyman   Join Date: Jan 2016 Posts: 313 Re: COTM: Basic Poker Math Thanks for writing this up, very good explainations!
 12-20-2017, 10:42 AM #25 TheStrumps enthusiast   Join Date: Nov 2017 Posts: 82 Re: COTM: Basic Poker Math 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. Last edited by TheStrumps; 12-20-2017 at 11:01 AM.

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