Micro Limit Math Thread

[Aaron W.]

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There's some truth to Shadow's comments about not doing math, however that really applies to playing at the table and only in the earlier stages of your development. When analyzing hands, you need to be able to pull out the numbers to show more decisively the values of each play.


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Emphasis provided.

Surely you mean "later stages of development"?

Anyway, I was sure to put a qualifier in there: "at table play". When analysing hands here, math is pretty essential. (I pretty much neglect it though [img]/images/graemlins/grin.gif[/img])

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No, I think the math becomes more important as you get better. Not necessarily math in the sense of recalculating the odds every single time, but I really believe that you need to be able to make quick and definite estimates when you sit on tough decisions.

My perspective has been undeniably altered by this old post which was recently revived in SS. Read everything GoT has to say. I think he's got it 100% correct when he talks about needing to put concrete numbers on things. Also read where The Dude quotes Mason's essay.

But this comes in at the later, because you can't put concrete numbers on a situation until you have enough experience to have the hand reading ability to start counting out the situations. Granted, in looser games, where villains tend to have huge hand ranges, this is more difficult. But even still, you'll find that if you pay attention you can put any player on a decent range of hands and compute your expectation from it. With experience, this comes faster and feels more natural. I don't do it very naturally just yet, but it's slowly coming along.

[marchron]

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To get the "to one" odds from a percentage divide one by the percentage and subtract one (to put on the other side of the ratio.)

25% = 1 / 25 = 4 = 3:1

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You either need to convert 25% into the decimal .25 or change the 1 into 100%.

[Dave G.]

Good post, I like the idea of the thread. I'm a final year maths student so I find all this stuff interesting.

I think the maths behind all of this is important for anyone hoping to advance beyond the beginner stages of the game. I just don't think it's possible to be as successful higher up without a solid maths foundation backing you up.

Anyway, I'll add this to the thread since I haven't seen it added yet. This is an idea I've seen Greg J post a few times, so you can give him credit for it - I'm just repeating it.

This is how to approximate your equity at the table. If there are two streets to come, multiply your estimated number of outs by 4. If there is one street to come, multiply your estimated number of outs by 2. This is only an approximation (and being a maths guy I hate approximations [img]/images/graemlins/tongue.gif[/img]) but it's fairly close.

Example: You flop a flush draw, so you have 9 outs (assuming only your flush outs give you the winning hand, which we will) with 2 streets to come. 4*9 = 36%. Your true equity, that is, the times you will complete your flush and win the hand, is 1 - (38/47 * 37/46) = 34.968%. So it's a little rough, but it's more than adequate, and it's a quick and easy calculation anyone can do it right at the table.

So yeah, if I think of other cool stuff to add, I'll post again, and I'm happy to help answering questions too if anyone wants to post any.

[Nfinity]

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Somebody else surely has questions. Or does everyone else in the micro-limits forum have it down pat and plays mathematically perfect poker?

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Most players are making mistakes that they don't recognize as mistakes. In the same way, most players are doing the math wrong, but don't even know it (or aren't even doing the math).

There's some truth to Shadow's comments about not doing math, however that really applies to playing at the table and only in the earlier stages of your development. When analyzing hands, you need to be able to pull out the numbers to show more decisively the values of each play.

As your hand reading goes up (when you have enough experience to really get hand reading), you can do quick hand counts and pot odds calculations to make better decisions.

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I think everyone should get a firm grip on the math from the word go. Your table skills and the analyzing you and your peers(us) do away from the table go hand in hand.

Lets say for example you post a hand. Something that is a fairly common situation. Now you get like 4 good replies, all of them saying to do something different, and all of them giving you a firm basis for the way the advise to play the hand. Now the next time you face this situation you are right back where you started. A lot of good topics in this forum end up this way. A whole lot of good discussion but no conclusive data.

Now say one of those posts gives you a strategy rife with mathematical explanation. These posts stick out in your mind, and will possibly lead you to take the correct course of action the next time.

That is assuming the poster did the math right. What could have happened was, he did it according to how he understands it(wrongly) and no body, including yourself caught it because no one decided to get a grip on the math.


Learning how to do the math helps you to better understand the advice you are given, you may even be able to figure out a lot more by yourself, saving your "forum time" for the more trickier things. It also helps us, as you will be able to better advise players and give more definitive advice(what this forum needs), and be able to catch those glaring errors that others post.

[Nfinity]

I'm glad you mentioned Dave G.

Here's another nugget of mathematical wisdom from that guy:

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Because I'm trying to improve my hand. If someone else flopped a set of 3's then they have 2 of my outs to improve. I know I have top set, but I also know I am behind a flush (95%) and need a boat to win.

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This is an incorrect way to view your outs. You don't know what your opponents have. In fact, it can be proven that this doesn't even matter; your odds don't change at all.

It's 4am and I'm in a mathematical mood, so I don't know if this post is going to be useful for anyone, but I feel like doing it anyway. So here goes.

I'll use the more simple example of a flush draw. The flop comes with 2[img]/images/graemlins/diamond.gif[/img]'s. You hold 2[img]/images/graemlins/diamond.gif[/img]'s, giving you a flush draw. There are 2 people in the pot against you (for simplicity I'll limit it to two; the calculations get too long for more than that).

There are 47 unseen cards, and 9 unaccounted for[img]/images/graemlins/diamond.gif[/img]'s. We say you have 9 outs. 9/47 = 0.191489 which is your probability of hitting a diamond on the turn.

With 2 other people, there's a decent chance that someone has one of your outs. Lets say we take this route, and assume that one of our opponents does indeed have a[img]/images/graemlins/diamond.gif[/img], and we discount our outs to 8. Once you do this, you are no longer dealing with a deck that has 47, or 46 unseen cards. That is, we don't just calculate 8/46 and that's our chance of hitting our diamond on the turn. We are considering the odds of all our opponents cards when we say that someone must have a diamond. So the new number of unseen cards, that is, the cards we are free to draw at, is 43. (2 opponents, 2 cards each).

Here's the correct way to do the new calculation. The probability that an opponent holds 0 diamonds is 38/47*37/46 = 0.6503. We'll call this p0. p2 (someone holds 2 diamonds) = 9/47*8/46 = .0333 and p1 = 1 - p0 - p2 = 0.3163.

Call m the probability of there being a certain number of total diamonds between our two opponents. If there are 0 diamonds between our opponents, there are still 9 diamonds left in our new 43 card deck, and the probability that neither opponent has one is equal to p0*p0. So,m0 = 9/43*p0*p0. (Neither opponent has a diamond).

Similarly,

m1 = 8/43*p0*p1 + 8/43*p1*p0. (Either player has a diamond, but the other does not).
m2 = 7/43*p0*p2 + 7/43*p1*p1 + 7/43*p2*p0 (Either one player can have two diamonds, or both players have one).
m3 = 6/43*p1*p2 + 6/43*p2*p1 (one player has one and the other has 2)
m4 = 5/43*p2*p2 (both players have 2 diamonds)

Now we have a list of probabilities. If we calculate m0+m1+m2+m3+m4, which is the total probability of all of these events, we get a probability of 0.191489. Which, as it turns out, is exactly the same as the initial probability with 47 unseen cards - 9/47.

It doesn't matter if you think your opponent has X card in his hand. Your odds of drawing do not change, because he will only have X card in his hand with some probability. And by considering that someone has that card in their hand, you have to take your opponents cards out of the deck when calculating odds.

[tor]

One thing that's been bugging me that I haven't really been able to find is a quick way to compute the different number of possibilities that an opponent can hold a given hand. Say...

Any two hearts
...with three hearts on the board.
Any heart
...with four hearts on the board.
Any Ace and Any King
...with an ace on the board
...with two aces and a king on the board
Two Aces
...with none on the board
...with one on the board
Any Ace
...with one known
...with two known

I know I could figure out all the combinations hand by hand, but I don't really trust that, plus it would be much easier (hopefully) if there were a set of quick equasions to do this with. Any help? Barring that, a chart somewhere? I didn't find one through my googling all yesterday.

[sirana]

</font><blockquote><font class="small">In Antwort auf:</font><hr />

Anyone want to symbolically integrate the following for me (with respect to x):

exp(1/S*sqrt(2*pi)*x*exp(-(ln(x-M))^2/(2*S^2)))

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hey wookie, you could have at least make yourselve some work and find something difficult

int(exp(1/S*sqrt(2*pi)*x*exp(-(ln(x-M))^2/(2*S^2))))

= 1/2S^2 * Int((exp(2*sqrt(2)*exp(ln(-M+x))*sqrt(pi)*x dx)/S

PS: what do you wanna say with symbolically?

[tor]

Anybody? Nobody here is interested in the math of the game, the huge WHY behind every decision?

[Aaron W.]

Computing outs and equity based on the chances of certain situations happening...

[shadow29]

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Anybody? Nobody here is interested in the math of the game, the huge WHY behind every decision?

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Not really.