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# Poker is totally mathematical

1. ## Poker is totally mathematical

I have just read a thread in which people said "poker is not all about maths," "luck is a factor in poker," "it is not always right to take +\$EV decisions."

I am now going to show that everything in poker can be expressed mathematically, and prove that those people who say poker is not all about maths are incorrect

Luck does not exist. Each poker "event" has a likelihood based on some measure of prevision or "probability" which is in turn based on your own judgements about the likely ranges of your opponents hands (from reads and bet sizing) and more deeply about your own judgements of the mechanics of the game.

If you believe in this mysterious "luck" idea then in order to be coherent you must alter your probability of winning a showdown. To make this concrete ill use an example from another thread:

I have AK and I believe my opponent has a pocket pair. He has gone all in after people limped and I had raised. I believe that the deck has been shuffled well so that as far as I can tell any card which I have not seen is equally likely to come down on the board. I have made a coherent probability statement so now I can work out my EV for the decision to call and the decision to fold, and take the decision with the maximum EV. In the thread the call was the right choice.

However believing in some mystical force called luck changes my statement that each card is equally likely to come out, and that in some way my aces and kings are not as likely/more likely to appear than the other cards. I must now respecify my probabilities and use those to work out my equity for the hand. If that turns out to be -\$EV then I should fold. But I fold ONLY because by believing in the mystical force luck I have changed the nature of the game in some sense.

But what of the argument that \$EV is not the only factor?

Well it is not! But all decision making is mathematical. You have a reward, which in the case of poker is money. You specify your utility for that reward, i.e. a measure of how much you prefer one amount over another amount. Then you make the decision that maximises your expected utility. This completely specifies the problem because not choosing this optimal decision means you didnt really believe your utilities in the first place and is therefore incoherent.

Clearly then \$EV is just a specific case of expected utility based on the assumption that we prefer, for example, \$2 twice as much as \$1. You can't just use \$EV all the time and think you are making the right decision though!! You have to be careful:

e.g. I have only \$100 in the world. Without it I can't eat and will starve for a whole week. Someone offers me to toss a coin for my \$100 and says he will give me \$210 if it is heads and moreover that I can use any fair coin and toss it myself (so that he has no influence on the game.)
My \$EV is clearly \$155 for playing and \$100 for not playing so based on that I should play. But would you knowing you'd starve if you lost????? In this case my Utility for money is not linear. I need my \$100 so badly that it dominates the function. Assymptotically, as my wealth increases and losing \$100 does not matter, then I always take the bet, as my utility tends to \$EV.

This has deep meaning for poker. If you are not properly rolled, then many decisions that are +\$EV are incorrect for you. Which is the mathematical reasoning behind playing with a proper bankroll, because if you do then your utility for money (in your roll) is proportional to the money itself and so \$EV decisions make sense.

So if you ever worry about the amount of money you lose if you make the call, (and this includes thinking that "later ill win more if i save this"), that is the same as playing scared and means that for you maximising \$EV is not a coherent way of making your decisions.

If maximising \$EV is not optimal for you, (and mathematically it really doesn't have to be and in many cases shouldn't be), then the decision making advice on this forum is incorrect for you. Only with a proper bankroll which must be treated as if infinite (in comparison to the stakes you play) so that your utility for money is completely linear, can the advice on here (and in nearly any book) make any sense.

If you are arguing for not taking an edge therefore, you are making a subjective statement equivalent to, "the money i'm risking here is too much for me" and thus posters on this forum can only tell you that you should stop playing the stakes you are if you want the mathematics of poker to make sense.

We would be wrong to say that you should always maximise youe \$EV, because as I have shown this may not make sense for you and your preferences about money. Clearly however we cannot give any advice to you either if your preferences are not completely specified to us. Furthermore if you do not subscribe to a linear utility function for money in your bankroll then you will not make as much money long term as someone who does.

If there is any decision in poker that is not mathematical, I challenge you to find it!
2. ## Re: Poker is totally mathematical

Great job Gingerwizard.

Originally Posted by gingerwizard
If maximising \$EV is not optimal for you, (and mathematically it really doesn't have to be and in many cases shouldn't be), then the decision making advice on this forum is incorrect for you. Only with a proper bankroll which must be treated as if infinite (in comparison to the stakes you play) so that your utility for money is completely linear, can the advice on here (and in nearly any book) make any sense.
QFT, and absolutely without argument

I will state that there are reasons to not take the +EV decisions in poker, but they are few and very specific. They revolve around tournements and sattelites. We won't get into specifics here, because the argument was not about the specific reasons for not taking +EV plays, but luck and EV in cash games,where reloading is an option and your poker life is never on the line.
3. ## Re: Poker is totally mathematical

Originally Posted by Trainer_jyms
Great job Gingerwizard.
thanks!

Originally Posted by gingerwizard
If maximising \$EV is not optimal for you, (and mathematically it really doesn't have to be and in many cases shouldn't be), then the decision making advice on this forum is incorrect for you. Only with a proper bankroll which must be treated as if infinite (in comparison to the stakes you play) so that your utility for money is completely linear, can the advice on here (and in nearly any book) make any sense.
QFT, and absolutely without argument

I will state that there are reasons to not take the +EV decisions in poker, but they are few and very specific. They revolve around tournements and sattelites. We won't get into specifics here, because the argument was not about the specific reasons for not taking +EV plays, but luck and EV in cash games,where reloading is an option and your poker life is never on the line.
Actually my friend you always take the +EV decisions! You must be careful however to make sure that what you are calculating is \$EV and not chip EV. +Chip EV can sometimes be illusory especially in SNGs and satellites as you correctly pointed out.
4.  02-21-2007 01:17 PM Halv pro crastinator Join Date Aug 2005 Posts 3,199 Location No hindsight for the blind. Great post. Now dissect these statements mathematically, they drowned in the Folding AK preflop thread when it descended into luck hell: Thread is a bit old but I believe all the posters stating that this is largely a "math" call have missed one important part. The fact that there were six callers lead to the not unreasonable conclusion that one or more of the outs are missing. People tend to play aces and kings so I don't think it unreasonable to expect two outs to be gone. Not sure what this does to the calculations. Obviously, you can't know this but they're not all holding pocket pairs. this is dumb What's wrong with this idea? It's in NLHETP. I'm sure Sklansky would believe one or two of the six is holding an ace or king. The fact that they're weak, live players only increases the odds of holding an ace. They always limp with Ax. Sure, they can limp with most anything, but aces and kings are good possibilities. Stop discussing "luck" and discuss this! Can one assign a % to the chance that someone folded one or more of our cards? At any rate, this % is probably negated by the % that he has a worse A himself or some random junk hand. I believe Bob Chiaffone says something similar in one of his books ("Improve your poker"?). First music vid: http://www.youtube.com/watch?v=MFerARdGW04 Free stream of different song here: http://www.nrk.no/urort/artist/wellfear ('Lytt'/play button on right side)
5.  02-21-2007 01:43 PM Trashcona Join Date May 2006 Posts 356 Great post, I spend a chunk of time every home game trying to explain to people that luck has nothing to do with poker with very little success. I'll have to point them to this post, very well put Gingerwizard.
6.  02-21-2007 01:46 PM jyms TILTING MOD Join Date Feb 2006 Posts 9,290 Originally Posted by Trashcona Great post, I spend a chunk of time every home game trying to explain to people that luck has nothing to do with poker with very little success. I'll have to point them to this post, very well put Gingerwizard. If you do that, they will read FTR, and there goes your "gravy train home game"
7.  02-21-2007 02:08 PM Da GOAT Join Date Feb 2006 Posts 4,923 Location Dublin Originally Posted by Trainer_jyms Originally Posted by Trashcona Great post, I spend a chunk of time every home game trying to explain to people that luck has nothing to do with poker with very little success. I'll have to point them to this post, very well put Gingerwizard. If you do that, they will read FTR, and there goes your "gravy train home game" i typed a longer post but im just gonna say this for home games. DO you tell your mates of your level of commitment to poker??? I dont, i know they are losing online (prob \$50 here and there, over time its alot) and in general. does this make me a poor mate by not advising them, tho im unsure they will listen (college mates but i dont party with them) Jman: every time the action is to you, it's an opportunity for you to make the perfect play.
8.  02-21-2007 02:17 PM jyms TILTING MOD Join Date Feb 2006 Posts 9,290 Depends on your definition of mates. guys I know, work with, or just play poker with, I tell them nothing. My Freinds, real freinds, they know my commitment, and think I'm full of shit, with how much I win and never deposit. They think that there is nothing to learn about poker, get your cards, bet or fold. That's what they play.
9.  02-21-2007 02:25 PM Halv pro crastinator Join Date Aug 2005 Posts 3,199 Location No hindsight for the blind. Meh, I tell anyone who asks how dedicated I am. I also point them to this forum. I don't really play my home game for profit only, it's more a social/recreational thing. I don't think any of them have taken my advice to go to the forums and study the game. Hell, my roommates play like a couple of hours online every day, and I've only just convinced them to get PT and PAHUD, they don't even bother posting hands in the forums. Bottom line is, to make money in this game takes dedication, and your average fish isn't going to have that dedication. First music vid: http://www.youtube.com/watch?v=MFerARdGW04 Free stream of different song here: http://www.nrk.no/urort/artist/wellfear ('Lytt'/play button on right side)
11.  02-21-2007 02:52 PM Halv pro crastinator Join Date Aug 2005 Posts 3,199 Location No hindsight for the blind. That was excactly what I was looking for. Thank you. First music vid: http://www.youtube.com/watch?v=MFerARdGW04 Free stream of different song here: http://www.nrk.no/urort/artist/wellfear ('Lytt'/play button on right side)
12.  02-21-2007 03:13 PM Da GOAT Join Date Feb 2006 Posts 4,923 Location Dublin Thx Halv. dont say much coz i want the games to be serious in nature. im sure they also wont bother with any advise etc as you say. Jman: every time the action is to you, it's an opportunity for you to make the perfect play.
13.  02-21-2007 03:59 PM IowaSkinsFan Join Date Oct 2004 Posts 8,895 I said this in the other thread but for those who believe in luck plz know that it's going to stunt your growth in poker. Check out the new blog!!!
14.  02-21-2007 08:26 PM donkbee SERIOUSLY WTF Join Date May 2005 Posts 6,261 Location outer space This is a great post. I'm gonna add it to the beginners digest in the BC. Thanks for writing it, gingerwizard. Originally Posted by Fnord Why poker fucks with our heads: it's the master that beats you for bringing in the paper, then gives you a milkbone for peeing on the carpet. blog: http://donkeybrainspoker.com/
15.  02-21-2007 09:12 PM bigslikk Join Date Apr 2006 Posts 446 Location Chicago "Gingerwizard's post" QFT QFT QFT
16.  02-21-2007 10:43 PM taipan168 Join Date Jul 2005 Posts 10,571 Location Sydney Superb post and very well explained and reasoned. I agree 100% - it's a great explanation of why playing with a sufficient BR is critical to this game.
17.  02-21-2007 10:46 PM Da GOAT Join Date Feb 2006 Posts 4,923 Location Dublin BTW nice post ginger Jman: every time the action is to you, it's an opportunity for you to make the perfect play.
19.  02-22-2007 05:52 AM ensign_lee Join Date Feb 2005 Posts 4,272 Location The University of TEXAS at Austin Aww. One of our posters is growing up. *sniffle*
20.  02-22-2007 08:01 AM Da GOAT Join Date Feb 2006 Posts 4,923 Location Dublin Originally Posted by ensign_lee Aww. One of our posters is growing up. *sniffle* Jman: every time the action is to you, it's an opportunity for you to make the perfect play.
22.  02-22-2007 02:06 PM Sprayed Join Date Nov 2005 Posts 3,326 Location GO BUCKS! You lost me at "Asymptotically". Great post gingerwizerd!
23.  02-22-2007 02:27 PM Da GOAT Join Date Feb 2006 Posts 4,923 Location Dublin Originally Posted by Sprayed You lost me at "Asymptotically". Great post gingerwizerd! Lost me at ''The''. Good post Ginger. Tho im now mentally unqualified to discuss further. I shall just nod in approval. nh sir Jman: every time the action is to you, it's an opportunity for you to make the perfect play.
24.  02-22-2007 02:33 PM gingerwizard Join Date Sep 2006 Posts 1,855 Originally Posted by Sprayed You lost me at "Asymptotically". Great post gingerwizerd! As everything tends to infinity This is not my signature. I just write this at the bottom of every post.
25.  02-22-2007 03:21 PM Sprayed Join Date Nov 2005 Posts 3,326 Location GO BUCKS! Originally Posted by gingerwizard Originally Posted by Sprayed You lost me at "Asymptotically". Great post gingerwizerd! As everything tends to infinity Yea, I had to look it up. Can you use it in a sentence please ? What is it's origin?
26.  02-22-2007 05:10 PM ensign_lee Join Date Feb 2005 Posts 4,272 Location The University of TEXAS at Austin From the word asymptote. Don't you remember high school math?
27.  02-22-2007 06:05 PM Sprayed Join Date Nov 2005 Posts 3,326 Location GO BUCKS! Originally Posted by ensign_lee From the word asymptote. Don't you remember high school math? I'm 32, I barely remember what happened last week.
29.  02-22-2007 08:41 PM Sci Fi Join Date Nov 2006 Posts 7 Pelion, Very good answer. Finally. I will say that I still believe the likelyhood that a greater percentage of aces and kings being played than on average is still not unreasonable. But the effect is clearly less than I had thought. Thanks.
32.  02-23-2007 04:16 AM jackvance Join Date Mar 2006 Posts 4,314 Instinct=your subconscious doing math. Sarcasm is your body's natural defense against stupidity
33.  02-23-2007 07:15 AM salsa4ever Join Date Jun 2005 Posts 1,363 Location Melbourne the challenge is of course not in the mathematical calculation, but in the estimation of the variables that go into the calculation Originally Posted by bigred Would you bone your cousins? Salsa would. Originally Posted by salsa4ever well courtie, since we're both clear, would you accept an invitation for some unprotected sex?
34.  02-23-2007 03:50 PM zook Join Date Mar 2006 Posts 5,034 Originally Posted by salsa4ever the challenge is of course not in the mathematical calculation, but in the estimation of the variables that go into the calculation QFMFT. This is why hand reading > > > math. The math is important, but if your pattern mapper's on the fritz, it doesn't do you much good.
35.  02-23-2007 04:03 PM Ash256 Join Date Mar 2006 Posts 2,883 Location UK Originally Posted by Chopper how bout when you know your opponent only plays the nuts and he bets 1/5 the pot on the river? sure, math says to call If your opponent has the nuts 100% of the time, math says you shouldn't call. If you have your opponent beat >20% of the time, math says to call. Amirite? Listen to my music
36.  02-23-2007 04:53 PM jackvance Join Date Mar 2006 Posts 4,314 Originally Posted by Ash256 Originally Posted by Chopper how bout when you know your opponent only plays the nuts and he bets 1/5 the pot on the river? sure, math says to call If your opponent has the nuts 100% of the time, math says you shouldn't call. If you have your opponent beat >20% of the time, math says to call. Amirite? Not really.. if he bets 1/5 pot you have to be good >14% (1/7) to make the call.. Sarcasm is your body's natural defense against stupidity
37.  02-25-2007 12:06 PM thenonsequitur Join Date Nov 2004 Posts 1,311 Location Location: Location Originally Posted by Chopper your win rates, or money earned, depend not only on mathematical probabilities, but also on your ability to possibly drop a hand when you know you are beat, but are still getting the odds to call. This is a logical contradiction. If you know you are beat, you are not getting odds to call, by definition.
38.  02-26-2007 11:12 AM gingerwizard Join Date Sep 2006 Posts 1,855 Originally Posted by zook Originally Posted by salsa4ever the challenge is of course not in the mathematical calculation, but in the estimation of the variables that go into the calculation QFMFT. This is why hand reading > > > math. The math is important, but if your pattern mapper's on the fritz, it doesn't do you much good. Hand reading is a pure skill. You make your decisions based on Math alone however. Math starts after hand reading as the hand reading specifies your probabilities to win. Getting 5 to 1 does not automatically make it a call. Only if you think you are better than 5 to 1 to win can you make a good call. If your hand reading sucks, then as zook says the math will not make you money. (But at least you make correct decisions based on the knowledge you have.) There is no excuse for deciding your probabilities, using that info to work out that you have a mathematically correct call, and then folding. Ever. This is not my signature. I just write this at the bottom of every post.
40. ## thanks!!

WOW!
thats pretty much all I have to say. that and THANKS
I am only now trying to learn a better game and your explanation surely helps. I hope to learn it,know it, do it.
41.  03-12-2007 02:51 AM bantam222 Join Date Dec 2005 Posts 224 Originally Posted by Trashcona Great post, I spend a chunk of time every home game trying to explain to people that luck has nothing to do with poker with very little success. I'll have to point them to this post, very well put Gingerwizard. Why do you have to prove this to them? One reason fish keep putting in money is that they think they are better then everyone else and the only reason they lose is because they are unlucky. Just keep them happy and keep taking their money because you "get lucky"
42.  03-12-2007 03:00 AM mrhappy333 Join Date Jan 2006 Posts 3,711 Location Hartford, CT Originally Posted by bantam222 Originally Posted by Trashcona Great post, I spend a chunk of time every home game trying to explain to people that luck has nothing to do with poker with very little success. I'll have to point them to this post, very well put Gingerwizard. Why do you have to prove this to them? One reason fish keep putting in money is that they think they are better then everyone else and the only reason they lose is because they are unlucky. Just keep them happy and keep taking their money because you "get lucky" qft, why school the fish, unless your all trying to get better. otherwise, keep taking their money, its a valuable lesson.
43.  11-23-2008 12:50 AM HarleyGuy13 Join Date Sep 2008 Posts 1,371 Location Oregon Originally Posted by mrhappy333 Originally Posted by bantam222 Originally Posted by Trashcona Great post, I spend a chunk of time every home game trying to explain to people that luck has nothing to do with poker with very little success. I'll have to point them to this post, very well put Gingerwizard. Why do you have to prove this to them? One reason fish keep putting in money is that they think they are better then everyone else and the only reason they lose is because they are unlucky. Just keep them happy and keep taking their money because you "get lucky" qft, why school the fish, unless your all trying to get better. otherwise, keep taking their money, its a valuable lesson. Outstanding post/topic. I must say I am in ahh of all your math skills for I am a basic fella. Was it Barry Greenstein who said "Don't tap on the glass, it scares the fish!"
44.  11-23-2008 10:05 AM Zel Join Date Oct 2008 Posts 60 Amazing Post.
45.  11-24-2008 07:23 AM Sugar Nut Join Date Jun 2008 Posts 44 Location grinderschool.com The following post is an attempt by me to describe psychology mathematically. I do know some poker math and I know it quite well. I am however not a math geek in any way. Everything I know about poker math was HARD WORK for me to understand. I am highly interested in math and have fun exploring it so I would appreciate if some math person could review my calculations and tell me where I screwed up. Thanks ------------------------------------------------------------------------------------------- Great post! There is something to be said about psychology, however, as it might in turn affect the math (as in: your EV) Let's say I get X-ray vision for one hand and can see my opponent's hole cards. He ships all in, i get x to y odds and can calculate my EV to the cent, which in this case happens to be \$1.34. Clearly I should call this based on math and my EV calc. However, if I know myself and my psyche well enough to know that losing this hand at this specific time might put me on monkey tilt, then an argument could be made to rather take the \$0.00EV decision than the \$1.34EV decision This situation can be described mathematically aswell, though as there is an x% chance for me losing the hand, thus a y% of x% = z% chance for me to go on monkey tilt after it, thus a z% chance for me to play -EV poker in the future. Now we could make a tilt-EV estimation based on empirical data (collect every hand in our DB that we played while monkey tilting and calculate the average EV of all our decisions ), then estimate the no. of hands that we will remain on monkey tilt and then calculate our true EV-call, which would look something like this I think: 1.34 - z% * [average no. of hands I tilt] * [average amount I lose per hand tilting] If the result is > 0 I should call, otherwise fold Example: I lose this hand 60% (x%) of the time and if I lose it there's a 50% (y%) chance for me to monkey tilt for an average of 100 hands, so the totaly chance for me to tilt when I call is 30% (z%). On average i lose \$0.10 per hand while tilting -1.66 = 1.34 - 0.3 * 100 * 0.1 So based on these assumptions we shold fold. Let's say we only tilt an average of 50 hands and lose only \$0.05 per hand tiltting: 0.59 = 1.34 - 0.3 * 50 * 0.05 Then we should call. Sugar Nut
46.  11-24-2008 07:29 AM Guest There's also psychology and math involved when you create people's ranges. What cards they have in their range is dictated by psychology, what your decision should be against his range is dictated by math. So your true EV depends on how close both of those calculations were. If you underestimate how tight the range is, you may call where a fold is better. If you overestimate someone's tightness you may fold where a call is better. If you mess up your math, you make a wrong decision as well. Since the math is easy to get right, it should be practiced until you can instantly say what % of the pot your call is, and what chance of winning you have. That will allow you to focus on the psychology part of it and try to understand what range someone is playing this way.
47.  11-24-2008 08:47 AM Da GOAT Join Date Feb 2006 Posts 4,923 Location Dublin ah whatever happened to gingerwizard Jman: every time the action is to you, it's an opportunity for you to make the perfect play.
48.  11-24-2008 12:45 PM renee_walmer Join Date Nov 2008 Posts 16 nice post

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