random numbers, large sets, and distribution

[hellokeith]

So I was playing roulette for a good hour or so, and I had gone way up early but was doing badly for a while. Add to that the 3-4 "free" drinks that I had, and it was apparent I needed to wrap up things. I noticed that #17 had not hit in a long, long time.. maybe even since I had sit down at the table.

Last spin. $5 chip on #17. Bingo! I won $175, walking away with about $125 profit.

So this gets me thinking. In general, if the numbers are evenly distributed to the point where one can predict that a particular number, due to its absence in x amount of outcomes, is more likely to occur, then that flies in the face of randomness.

Put another way, a truly random causality dictates that it is possible for #17 to hit 25 times in a row, or, not hit in 3500 events. We intuitively laugh at either prospect, because we, for lack of a better term, believe randomness means, given a large enough set of events, even number distribution.

So which part is flawed? The part that assumes roulette spins are random, or the part that assumes the numbers will be evenly distributed over a set "sufficiently large" to qualify as random?

 

[TuxDave]

I think you just got lucky. Assuming roulette has an even distribution of independent events, you can never claim that a number has a higher probability of occuring just because it hasn't showed up in a while.

 

[Aluvus]

Originally posted by: hellokeith
So this gets me thinking. In general, if the numbers are evenly distributed to the point where one can predict that a particular number, due to its absence in x amount of outcomes, is more likely to occur, then that flies in the face of randomness.

You can't make that prediction. Each spin of the roulette wheel is random and independent, with an equal probability for each of the possible outcomes*, and whether 17 has "hit" recently or not has no bearing on whether it will hit on the next spin. This is a classic example of gambler's fallacy.

You just got lucky. I hope you were at least playing at a table without the 00.


* It is possible for the physical wheel to introduce some sort of bias, but a huge amount of money goes into ensuring that is not the case.

 

[hellokeith]

The mistake both of you are making, hence this thread and my proposition, is that even distribution, regardless how large is the number set, flies in the face of true randomness. We suppose random means "lots of different numbers", but that is incorrect. Random means "any number, any time", which predicts #17 happening 140000 times consecutively, with no guarantee it will continue or won't continue.

 

[Tony Eveready]

double post

 

[CycloWizard]

Lucky bastard. My one time in Vegas, I was up $50 until it hit 27 three times in a row. Then I was down $100.

I think the problem with your approach is that you took this one isolated event as proof of your hypothesis - that 17 was any more likely than any other number. This is simply false. 17 was just as likely as any other number, theoretically. A roulette table can be thought of as a modified von Mises distribution with infinite variance. Since there are a finite number of values, each of which is equally likely (again, assuming an ideal design), there should be no mean value. A von Mises distribution doesn't have infinite tails like a Gaussian distribution - it wraps around and is continuous across the entire domain. Having infinite variance implies that there is no mean and, therefore, no likelihood of any point being hit more often than any other given infinite samples. Given infinite samples, hitting #17 140,000 times in a row will become insignificant.

Hopefully that makes sense. It's been a long time since I did anything with von Mises distributions and it's really late. I'm not even sure that's what it's called anymore.

 

[StopSign]

I think you're confusing randomness with probability. Given that you have a rough idea of how the distribution looks for the roulette wheel while playing, you knew, based on the distribution, that #17 had a "fairly good" chance of coming up because it has not not occurred yet. In order to fit the distribution of the wheel, #17 will eventually come up.

To answer the question, I think the first case is flawed because the roulette wheel might not be completely random. Where it lands could depend on a lot of things. If all conditions are controlled you can potentially replicate the spin over and over. There distribution is a result of the fact that it is nearly impossible to control the conditions and therefore no spin is replicated for a large number of spins.

 

[f95toli]

Originally posted by: StopSign
Given that you have a rough idea of how the distribution looks for the roulette wheel while playing, you knew, based on the distribution, that #17 had a "fairly good" chance of coming up because it has not not occurred yet.

NO, whether or not #17 had already occured has absolutely nothing to do with it; this is -as has already been pointed out- just gambler's fallacy.

The fact that a disitbitution is even simply means that every number occurs with the same probability. #17 200 times in a row is just as likely as another serier of numbers.

It is interesting to note that one method frequently used to check whether or not someone is cheating is to check whether or not e.g. the number #17 DOES occur 10 times in a row; given a large enough sample this is in fact likely to happen; but since we "intutively" think that such series look "non-random" cheaters tend to avoid them.

 

[TuxDave]

Originally posted by: hellokeith
The mistake both of you are making, hence this thread and my proposition, is that even distribution, regardless how large is the number set, flies in the face of true randomness. We suppose random means "lots of different numbers", but that is incorrect. Random means "any number, any time", which predicts #17 happening 140000 times consecutively, with no guarantee it will continue or won't continue.

So remind me how I'm wrong again? The probability distribution is known but the event is still random. Where am I wrong?

 

[sjwaste]

Originally posted by: f95toli

Originally posted by: StopSign
Given that you have a rough idea of how the distribution looks for the roulette wheel while playing, you knew, based on the distribution, that #17 had a "fairly good" chance of coming up because it has not not occurred yet.

NO, whether or not #17 had already occured has absolutely nothing to do with it; this is -as has already been pointed out- just gambler's fallacy.

The fact that a disitbitution is even simply means that every number occurs with the same probability. #17 200 times in a row is just as likely as another serier of numbers.

It is interesting to note that one method frequently used to check whether or not someone is cheating is to check whether or not e.g. the number #17 DOES occur 10 times in a row; given a large enough sample this is in fact likely to happen; but since we "intutively" think that such series look "non-random" cheaters tend to avoid them.

You mean, if 17 came out 1000 times in a row, each occurrence was just as likely as any other number to come out.

The probability of 17 coming out on each of a series of 1000 spins is much greater than the probability of each of 1000 spins being exactly 17.

If, say, the wheel has 1-36, 0, and 00, then there's a 1/38 chance on each spin that it will be 17. However, the probability of 17 coming out 1000 times in a row is (1/38)^17, if I'm not mistaken.

Or I could be way off...

 

[silverpig]

Originally posted by: sjwaste
You mean, if 17 came out 1000 times in a row, each occurrence was just as likely as any other number to come out.

The probability of 17 coming out on each of a series of 1000 spins is much greater than the probability of each of 1000 spins being exactly 17.

If, say, the wheel has 1-36, 0, and 00, then there's a 1/38 chance on each spin that it will be 17. However, the probability of 17 coming out 1000 times in a row is (1/38)^17, if I'm not mistaken.

Or I could be way off...

That's right.

Here's a question for you guys that might help illustrate something.


What sequence is more likely in roulette:

17, 17, 17, 17, 17, 17, 17
11, 19, 22, 14, 6, 26, 30



The answer is they are equally likely.

 

[hellokeith]

So then, true randomness does not mandate even distribution. The question then becomes, was the roulette wheel designed for true randomness or even number distribution?

 

[CycloWizard]

Originally posted by: hellokeith
So then, true randomness does not mandate even distribution. The question then becomes, was the roulette wheel designed for true randomness or even number distribution?

True randomness does mandate an even distribution for infinitely many samples. I think this is the part that is missing.

 

[silverpig]

Originally posted by: hellokeith
So then, true randomness does not mandate even distribution. The question then becomes, was the roulette wheel designed for true randomness or even number distribution?

Both. If the slots on the wheel were different sizes, or if there were two #17s then it would be neither.

 

[TuxDave]

Originally posted by: hellokeith
So then, true randomness does not mandate even distribution. The question then becomes, was the roulette wheel designed for true randomness or even number distribution?

If 'true randomness' means any number is equally likely to appear next and 'even number distribution' means any number is equally likely to appear next, then the ideal roulette table is designed for both.

If 'even number distribution' means every number appears exactly an equal number of times, then no, a random event with a even probability distribution function like a roulette does not mandate that every number MUST appear and eqaul number of times all the time.

An ideal roulette table is designed for 'true randomness' assuming the definition stated above.

 

[f95toli]

Originally posted by: CycloWizard

Originally posted by: hellokeith
So then, true randomness does not mandate even distribution. The question then becomes, was the roulette wheel designed for true randomness or even number distribution?

True randomness does mandate an even distribution for infinitely many samples. I think this is the part that is missing.

CycloWizard is right. In theory you need an infinite number of events to get a "perfect" distribution.
In experiments where we are interested in properties of a distribution (e.g. a gaussian distribution) we typically need a very large number of events; typically 10-20 000 is enough to allow us to infer certain properties. However, even then the distribution is far from "perfect" (i.e. it doesn't exactly follow the theoretical curve)

 

[CSMR]

Originally posted by: hellokeith
So this gets me thinking. In general, if the numbers are evenly distributed to the point where one can predict that a particular number, due to its absence in x amount of outcomes, is more likely to occur, then that flies in the face of randomness.

No it flies in the face of independence.

Put another way, a truly random causality dictates that it is possible for #17 to hit 25 times in a row, or, not hit in 3500 events. We intuitively laugh at either prospect, because we, for lack of a better term, believe randomness means, given a large enough set of events, even number distribution.

Are you many people posting under one username?

 

[CSMR]

Originally posted by: hellokeithRandom means "any number, any time", which predicts #17 happening 140000 times consecutively, with no guarantee it will continue or won't continue.

http://mathworld.wolfram.com/RandomVariable.html

 

[TuxDave]

Originally posted by: f95toli

Originally posted by: CycloWizard

Originally posted by: hellokeith
So then, true randomness does not mandate even distribution. The question then becomes, was the roulette wheel designed for true randomness or even number distribution?

True randomness does mandate an even distribution for infinitely many samples. I think this is the part that is missing.

CycloWizard is right. In theory you need an infinite number of events to get a "perfect" distribution.
In experiments where we are interested in properties of a distribution (e.g. a gaussian distribution) we typically need a very large number of events; typically 10-20 000 is enough to allow us to infer certain properties. However, even then the distribution is far from "perfect" (i.e. it doesn't exactly follow the theoretical curve)

I'm not so sure that the probability of getting a perfect distribution goes to 1 as your number of samples goes to infinity. Look at a case of a coin toss where the perfect distribution is half heads and half tails.

Given 2n tosses, the total combinations where exactly half are heads is (2n)!/(n!*n!)

The total number of possible results is 2^2n.

Therefore the probability of exactly half heads is 2n!/[n!*n!*2^2n] and as n goes to infinitely the probability goes towards 0. (ok I'm not 100% sure it goes to zero but it looks like it)

 

[f95toli]

Originally posted by: TuxDave
I'm not so sure that the probability of getting a perfect distribution goes to 1 as your number of samples goes to infinity. Look at a case of a coin toss where the perfect distribution is half heads and half tails.

Given 2n tosses, the total combinations where exactly half are heads is (2n)!/(n!*n!)

The total number of possible results is 2^2n.

Therefore the probability of exactly half heads is 2n!/[n!*n!*2^2n] and as n goes to infinitely the probability goes towards 0. (ok I'm not 100% sure it goes to zero but it looks like it)

You could very well be right, what I had in mind was continous distributions (e.g Gaussians).
However, if you instead calculate the relative populations I think you would find they go towards 50:50 as the number of events goes to infinity (meaning the mean goes towards exactly 0.5).

 

[CSMR]

The empirical distribution (proportion below a certain point) will tend to the probability distribution if the samples are independent. That doesn't happen in all states of the world just it happens with probability 1. It doesn't depend on the probability distribution.

 

[CycloWizard]

Originally posted by: TuxDave
I'm not so sure that the probability of getting a perfect distribution goes to 1 as your number of samples goes to infinity. Look at a case of a coin toss where the perfect distribution is half heads and half tails.

Given 2n tosses, the total combinations where exactly half are heads is (2n)!/(n!*n!)

The total number of possible results is 2^2n.

Therefore the probability of exactly half heads is 2n!/[n!*n!*2^2n] and as n goes to infinitely the probability goes towards 0. (ok I'm not 100% sure it goes to zero but it looks like it)

I'm a little rusty on my probability, but I think your problem would be governed by a Poisson distribution, not a Gaussian distribution. IIRC, the Poisson distribution governs discrete events (such as a coin flip), while the Gaussian distribution governs continuous responses. I could be wrong though... It's been a long time since I read up on such things.

 

[TuxDave]

Originally posted by: CycloWizard

Originally posted by: TuxDave
I'm not so sure that the probability of getting a perfect distribution goes to 1 as your number of samples goes to infinity. Look at a case of a coin toss where the perfect distribution is half heads and half tails.

Given 2n tosses, the total combinations where exactly half are heads is (2n)!/(n!*n!)

The total number of possible results is 2^2n.

Therefore the probability of exactly half heads is 2n!/[n!*n!*2^2n] and as n goes to infinitely the probability goes towards 0. (ok I'm not 100% sure it goes to zero but it looks like it)

I'm a little rusty on my probability, but I think your problem would be governed by a Poisson distribution, not a Gaussian distribution. IIRC, the Poisson distribution governs discrete events (such as a coin flip), while the Gaussian distribution governs continuous responses. I could be wrong though... It's been a long time since I read up on such things.

I'm just doing: Limit as n goes to infinity of

[number of situations of n heads in 2n tosses]/[total number of possibilities].

to prove that as you go to an infinite number of tosses, the probability of getting n heads out of 2n coin tosses does not go to 1.

I'm not clear on where and how I'm using an incorrect distribution type.

 

[imported_iNGEN]

An even distrobution of population is a presumption used for statistical analysis of sampled mutually exclusive events. In other words, even distrobution is a known error! There is no valid reason to believe that the population of roulette spins is, or approximates, even distrobution. As f95toli showed us, the probability of even distrobution actually decreases with population size.

Your mind is just trying to justify Gambler's Fallacy, because you observed a single coincidental occurence.

 

[Estrella]

I know that being human when we think of random, we tend to think of it as being evenly distriubted.