Big Numbers and Bad Runs

[Erich Schulte]

Question for the mathies. How does the law of big numbers apply to runs
of cards in poker? I understand that it would be impossible to have a
very bad run of coin tosses over any leanght of time. What is the
relevant difference with poker?

Is it that there are so many more variables? People can fold to your
big hands, call your bluffs, draw out on you, you can have a monster
agaist a bigger monster and when all of these things happen at once, you
have a bad run?

But it seems like there are a lot of varaibles that go into a real coin
toss as well, although more mundane ones. And there are certainly more
variables in something like a political poll. If a poll of 1400 people
can come pretty close to representing the views of America, why are my
last 1500-2000 hands so horrifying, with pretty minimal tilting? (yes, of
course I'm on a bad run myself.)

When you add up all of the variables, let's say a good player has an
expected return of $1.05 per dollar bet... or whatever number you like.
When does sample size overwhelm chace?

I'll throw this out too. Krieger had a column recently in which he
talked about simulating 30 years worth of hands for identical playing
styles and opponants and still found a significant difference in outcome.
1-2% I believe.

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[whatever]

You are mistaken about not having bad runs of coin tosses over time. The
laws of probability state that on EACH coin flip, the chances of hitting
either side are 50/50, however there is nothing that states over the long
run one side won't win out over the other, even by a long margin.

Many years ago, back when I was in college, we did an experiment on this in
a probability class - 300 lab students pitching quarters an hour a day,
everyday for 2 weeks, and in not one case did anyone ever come out even or
close to even. Either heads or tails won by a large margin. Odds are just
that - odds. They give you probability - but reality can often differ from
theoretical probability.


"Garycarson1" <garycarson1@wmconnect.com> wrote in message
news:20030927190844.24184.00000190@mb-m11.wmconnect.com...
> >Question for the mathies. How does the law of big numbers apply to runs
> >of cards in poker? I understand that it would be impossible to have a
> >very bad run of coin tosses over any leanght of time. What is the
> >relevant difference with poker?
>
> It's the Law of Large Numbers.
>
> Really all it says is that infinity is such a long time that the effect of
any
> run of finite length (even a bery very long run) is going to be washed
away by
> the rest of initinity.
>
> It doesn't say you don't have runs of large duration. It says that runs
of
> large duration don't have a meaningful effect on the long run average.

[TD Lowball]

On Sep 27 2003 3:45AM, Erich Schulte wrote:

> Question for the mathies. How does the law of big numbers apply to runs
> of cards in poker? I understand that it would be impossible to have a
> very bad run of coin tosses over any leanght of time. What is the
> relevant difference with poker?

Impossible or improbable?

> Is it that there are so many more variables? People can fold to your
> big hands, call your bluffs, draw out on you, you can have a monster
> agaist a bigger monster and when all of these things happen at once, you
> have a bad run?

Everyone gets the same distribution of cards in each postion over the long
run. It's what you do with them that counts.

> > But it seems like there are a lot of varaibles that go into a real coin
> toss as well, although more mundane ones. And there are certainly more
> variables in something like a political poll. If a poll of 1400 people
> can come pretty close to representing the views of America, why are my
> last 1500-2000 hands so horrifying, with pretty minimal tilting? (yes, of
> course I'm on a bad run myself.)

Becuase a poll is a sampling of a larger population, and there are
statisical forumlas for estimating the significance of the results. Also
polls are notoriously unrepresentative.

Poker hands need to be thought of as an endelss series of discrete random
events. Is anything I think poker hands could be represented as the sum of
several poission distributions

> When you add up all of the variables, let's say a good player has an
> expected return of $1.05 per dollar bet... or whatever number you like.
> When does sample size overwhelm chace?

Generally the absolute minimum should be 50 hands.

> I'll throw this out too. Krieger had a column recently in which he
> talked about simulating 30 years worth of hands for identical playing
> styles and opponants and still found a significant difference in outcome.
> 1-2% I believe.

Thats to be expected. Here is a quick example.

+++
Minitab 13.0 , two sets of 5000 random numbers, generated to be Normally
distributed with mean = 0, STDDEV = 1.0. These datasets should be nearly
equivalent.


Variable N Mean Median TrMean StDev SE
Mean
C1 5000 -0.0302 -0.0350 -0.0269 1.0040
0.0142
C2 5000 0.0210 0.0231 0.0188 0.9959
0.0141
+++

Already we see that player one has had a slightly worse set of hands

A quick Mann Whitney U Test (Testing if the population median's are
different, this test makes no assumptions about the distribution of the
underlying data, but needs large data sets to give meaningful results.).

+++
Mann-Whitney Test and CI: C1, C2

C1 N =5000 Median = -0.03497
C2 N =5000 Median = 0.02308
Point estimate for ETA1-ETA2 is -0.04591
95.0 Percent CI for ETA1-ETA2 is (-0.08589,-0.00593)
W = 24677981.5
Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at 0.0246
The test is significant at 0.0246 (adjusted for ties)
+++

So we see that with 5000 samples There can be slight and statistically
signifigant variations. I chose 5000 samples which is reperesentative of 1
month of heavy online play (50 hours at 100hands/hr).

TD Lowball --

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