Why does probability work?

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Re: Why does probability work?

#21  Postby Thommo » Nov 18, 2018 7:39 pm

Evolving wrote:On the bell curve, I interpreted the statement as referring to a large number of experiments, each consisting of a certain number of coin flips: the results of the experiments would cluster around 50-50 in the shape of a normal distribution.


Well, it would certainly be a sensible question to ask why a normal distribution is a good approximation for a binomial distribution for large N. The answer is probably going to come in the form of some algebra, which will follow again from a series of assumptions (I seem to recall that you need larger sample sizes for more skewed binomial distributions).
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Re: Why does probability work?

#22  Postby Thommo » Nov 18, 2018 7:43 pm

BWE wrote:
Thommo wrote:
BWE wrote:
surreptitious57 wrote:Probability works because it attempts to model reality based on observation
Every probability exists between 0 and I and how accurate a prediction will
be is determined by the accuracy and completeness of the data in question

You toss a coin I00 times probability says it will land heads 50 times and tails 50 times
Reality however says it is 5I / 49 because one side is marginally heavier than the other

But these are all tautological answers. My question is why does the coin flip vary around a bell curve? Why wouldn't it be always heads for no reason at all?


Coin flips are not distributed by a bell curve (AKA Normal distribution or Gaussian distribution), they are binomial distributions.

There's an important theorem in mathematics that governs the distribution of sample means from a wide range of arbitrary distributions, known as the https://en.wikipedia.org/wiki/Central_limit_theorem, which says that for large sample sizes the distribution of sample means is approximately normal regardless of the shape of the original distributions.


I think you understood my point though.


Honestly, not really. I can't discern any difference between the first question, for example, and the question "Why does counting work?" with the assertion "There is no reason it should.".

It's very difficult to tell what the question is after, and the statement makes no sense to me at all, I can't see what it's founded on.
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Re: Why does probability work?

#23  Postby newolder » Nov 18, 2018 7:49 pm

The required steps to get a Gaussian from a large N "fair" binomial distribution are laid out by Jake Hofman here.
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Re: Why does probability work?

#24  Postby Thommo » Nov 18, 2018 7:54 pm

newolder wrote:The required steps to get a Gaussian from a large N "fair" binomial distribution are laid out by Jake Hofman here.


Nice link, just what I was looking for! :thumbup:
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Re: Why does probability work?

#25  Postby BWE » Nov 18, 2018 8:57 pm

Evolving wrote:On the bell curve, I interpreted the statement as referring to a large number of experiments, each consisting of a certain number of coin flips: the results of the experiments would cluster around 50-50 in the shape of a normal distribution.

Yeah.
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Re: Why does probability work?

#26  Postby BWE » Nov 18, 2018 8:59 pm

Thommo wrote:
BWE wrote:
Thommo wrote:
BWE wrote:
But these are all tautological answers. My question is why does the coin flip vary around a bell curve? Why wouldn't it be always heads for no reason at all?


Coin flips are not distributed by a bell curve (AKA Normal distribution or Gaussian distribution), they are binomial distributions.

There's an important theorem in mathematics that governs the distribution of sample means from a wide range of arbitrary distributions, known as the https://en.wikipedia.org/wiki/Central_limit_theorem, which says that for large sample sizes the distribution of sample means is approximately normal regardless of the shape of the original distributions.


I think you understood my point though.


Honestly, not really. I can't discern any difference between the first question, for example, and the question "Why does counting work?" with the assertion "There is no reason it should.".

It's very difficult to tell what the question is after, and the statement makes no sense to me at all, I can't see what it's founded on.

It's more than why does counting work but it may have the same root issue. With the coin flip the question is why does the average hover around 50%? Maybe it's feynman's why do magnets work issue but I don't think so.
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Re: Why does probability work?

#27  Postby Blackadder » Nov 18, 2018 9:23 pm

BWE wrote:
It's more than why does counting work but it may have the same root issue. With the coin flip the question is why does the average hover around 50%? Maybe it's feynman's why do magnets work issue but I don't think so.


Because the mathematics (probability) is describing a physical system that has only two states and which has no systemic bias towards one or the other state. If you are asking why such a physical system behaves as it does, I'm not sure you will get an answer any more profound than "it just does". If the physical world were to produce results different to those we actually experience, e.g. a coin toss always came up heads, then probability theory would have to be adapted to describe THAT reality instead of the one we currently observe.
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Re: Why does probability work?

#28  Postby TopCat » Nov 18, 2018 9:37 pm

BWE wrote:With the coin flip the question is why does the average hover around 50%?

Let me try a partial explanation. I expect someone else will be able to improve on it.

Flipping a coin is not random like radioactive decay is random.

When you flip a coin, you apply a torque, it spins the coin, the kinetic energy dissipates against air resistance, and then it stops spinning. If you could do that exactly the same every time, in still air, it would stop in the same way every time.

But you can't reliably apply the same flip force every time. Every time, the force is a little different, and tiny differences make the difference between it stopping on H or T. It's no more likely that you'll add a little more force next time, and it'll stop on T rather than H, or a little less, or the same, or whatever.

The difference needed to change the outcome is way smaller than the amount of control you can exercise, so the outcome seems random.

Contrast coin flips with knife flips. Table knives, that is - they aren't sharp enough to cut myself if I catch it with the blade. If I'm flipping one, I can reliably flip a half rotation 100% of the time. I can flip a whole rotation about 95% of the time. Two rotations, about 70% of the time, but the rest is either 1.5 or 2.5 rotations. The point is, I can control the flip force well enough to predict the outcome, so it's not random.

Does this help?
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Re: Why does probability work?

#29  Postby BWE » Nov 19, 2018 1:54 am

Maybe. Radioactive decay might be a better example but you gave me some good ideas in there.
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Re: Why does probability work?

#30  Postby Thommo » Nov 19, 2018 3:04 am

BWE wrote:
Thommo wrote:
BWE wrote:
Thommo wrote:

Coin flips are not distributed by a bell curve (AKA Normal distribution or Gaussian distribution), they are binomial distributions.

There's an important theorem in mathematics that governs the distribution of sample means from a wide range of arbitrary distributions, known as the https://en.wikipedia.org/wiki/Central_limit_theorem, which says that for large sample sizes the distribution of sample means is approximately normal regardless of the shape of the original distributions.


I think you understood my point though.


Honestly, not really. I can't discern any difference between the first question, for example, and the question "Why does counting work?" with the assertion "There is no reason it should.".

It's very difficult to tell what the question is after, and the statement makes no sense to me at all, I can't see what it's founded on.

It's more than why does counting work but it may have the same root issue. With the coin flip the question is why does the average hover around 50%? Maybe it's feynman's why do magnets work issue but I don't think so.


Symmetry.

The universe is indifferent to objects that have indifferent properties.

If you make a "coin" that instead of being a flat disc with two identical faces is instead a hemisphere you will find it does not land heads up 50% of the time. Or if you just take a standard coin and bend it a few degrees you will find it does not land heads up 50% of the time.





-----

To answer in more depth:

It makes little sense to ask why coins land heads up 50% of the time and relate that to probability theory for a couple of reasons:

  1. The mechanics of a coin flip are well known, and essentially deterministic. Given a fixed initial upward force (at specificed g, air pressure and assuming no wind speed and that it lands on a fairly regular surface, all of which are realistic assumptions for some conditions) and initial upwards rotational speed it is possible to deterministically predict how many half-rotations take place before the coin rests, and thus which side is face up when it rests.

  2. The shape of an approximately fair coin is such that it has two large stable bases, which are labelled "heads" and "tails". The properties of these stable bases are roughly symmetric, meaning that if the coin contacts the ground below a certain critical downward speed and with below a certain critical rotational speed (that both vary with the angle of the coin at the point of impact) it does not have enough energy to escape the stable state of resting with the side that is currently closest to down, down.

  3. Those coin flips which are conducted with an arbitrary and unknown side face up have already undergone a pseudo-randomisation. A complex set of unpredictable factors will lead to you taking the coin from your pocket with a specific side up.

  4. Above a certain rotational speed and initial upward velocity the system becomes highly sensitive to initial conditions.

  5. Those shapes which do not have these properties, typically because they have an asymmetry of density distribution or shape do not have the property of landing heads up 50% of the time. Those times the flip itself is of controlled fashion (e.g. people who have practised throwing the coin upward with close to zero rotational speed) do not have the property of landing heads up 50% of the time.

  6. Probability theory does not only apply to events with p = 0.5. Events with different probability are covered by different distributions.

Examples:
A shape that has physical properties (and uniform density), such that it always lands with a particular side up, a long way from 50% heads and 50% tails!:


An approximately fair and symmetric object subjected to approximately fixed initial conditions will land with a large bias of one side being face down:
https://en.wikipedia.org/wiki/Buttered_toast_phenomenon
In the past, this has often been considered just a pessimistic belief. A study by the BBC's television series Q.E.D. found that when toast is thrown in the air, it lands butter-side down just one-half of the time (as would be predicted by chance).[4] However, several scientific studies have found that when toast is dropped from a table (as opposed to being thrown in the air), it does fall butter-side down.[5][6][7] A study by Robert A J Matthews won the Ig Nobel Prize in 1996.[8][9]

http://iopscience.iop.org/article/10.10 ... 4/005/meta
The orthodox view, in contrast, is that the phenomenon is essentially random, with a 50/50 split of possible outcomes. We show that toast does indeed have an inherent tendency to land butter-side down for a wide range of conditions. Furthermore, we show that this outcome is ultimately ascribable to the values of the fundamental constants.



I'm just going to round out the post (which has ended up far longer than intended), with the single word that sums the whole thing up, because it's incredibly important physically and mathematically. So much so that I can't really wrap my head around why anyone would question why laws of physics could act differently on identical states. I almost posted my reply after writing that single word.

Symmetry.
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Re: Why does probability work?

#31  Postby Thommo » Nov 19, 2018 3:41 am

There's also a lot that could be said about the difficulty of creating randomisation to an arbitrarily high level of precision, the importance of the variation of distribution parameters;
Image
and how systemic biases can be exploited.

There are very famous cases of both blackjack and roulette where additional information allows probability distributions to vary a lot from the assumption:
https://www.newscientist.com/article/dn ... to-set-up/
Packard should know. In the late 1970s, he and a group of other physics graduate students set out to create computers that could compute the sector of the wheel a roulette ball would land in. They hid these computers under their clothes or in their shoes, clicking buttons with their toes.

“In the best circumstances, we could predict the quadrant correctly,” says Packard. “We definitely got to the point where we were winning money, but we didn’t continue long enough to make large amounts.”


https://en.wikipedia.org/wiki/MIT_Blackjack_Team
The MIT Blackjack Team was a group of students and ex-students from Massachusetts Institute of Technology, Harvard Business School, Harvard University, and other leading colleges who used card counting techniques and more sophisticated strategies to beat casinos at blackjack worldwide. The team and its successors operated successfully from 1979 through the beginning of the 21st century. Many other blackjack teams have been formed around the world with the goal of beating the casinos.


In the world of computing there's a lot to be said about the roles of PRNG and the ways it can be detected, for example:
http://www-users.math.umn.edu/~garrett/ ... /pRNGs.pdf
Or an analysis I once read about a game I played:
http://www.schwanenlied.me/yawning/XCOM/XCOMPRNG.html
Given the nature of the algorithm, running any in-depth battery of tests against it is sort of a futile gesture. By the nature of the algorithm it uses, any stringent test suite will expose the fact that the LCGs just aren't that great.

That serves as an interesting introduction to the idea of Linear congruential generators and how strings of results can differ from random strings, via things like serial correlations even while individual results might fit a uniform distribution in their specified range.
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