Status
Not open for further replies.

#### MrChips

Joined Oct 2, 2009
31,087
I don't see where is the ambiguity. We keep going around in circles. You ask the same question. We give you the same answer.

After a number of n1 tosses, you get one result, H1 : (n1-H1).

After another number of n2 tosses, you get another result, H2 : (n2-H2).

It does not matter which side H or T has a lead for each experiment.
When you do another trial of 1000,000,000 tosses AND include the results of the previous n1 and n2 tosses, the results H1 and H2 are buried in the noise of the statistical data.

#### BobTPH

Joined Jun 5, 2013
9,278
and we agree that after a sufficiently large number of tosses (B), then there will be the same number of H as T.
No, we do not agree to that. In fact, the probability of getting an equal number of heads and tails goes DOWN as the number of trials goes up.

Here are the probabilities of getting in equal number of H and T for 10, 100, 1000, and 10000 trials:

10 0.24
100 0.08
1000 0.025
10000 0.008

Bob

Last edited:

#### ErnieM

Joined Apr 24, 2011
8,382
...Given 999 tosses,
say we have 490 H and 509 Tails. We can find the probability of that.

...So over the next "n" samples there must be more H than T to move towards the 50:50,
if there were not more H, then T would dominate, and the coin is not fair
One must realize that flipping another 999 tosses and getting 490 Tails and 509 Heads (that brings us back to DO do do do do) is equally likely/unlikely as the initial set of tosses.

#### djsfantasi

Joined Apr 11, 2010
9,188
No, we do not agree to that. In fact, the probability of getting an equal number of heads and tails goes DOWN as the number of trials goes up.

Here are the probabilities of getting in equal number of H and T for 10, 100, 1000, and 10000 trials:

10 0.24
100 0.08
1000 0.025
10000 0.008

Bob
It’s actually very simple. It is more likely that you WON’T get a 50:50 split. What you’ll experience are numbers CLOSE to a 50:50 split. How close depends on the total number of flips, but will ALMOST NEVER be a 50:50 split.

#### SamR

Joined Mar 19, 2019
5,081
You may also want to look into the standard distribution bell curve for probability.

Last edited:

#### BobTPH

Joined Jun 5, 2013
9,278
It’s actually very simple. It is more likely that you WON’T get a 50:50 split.
Not only that, but, the more trials, the less likely it is to get 50-50. The probability of getting an equal number peaks at 2 trials at 50% and only goes down from there.

Bob

#### Deleted member 115935

Joined Dec 31, 1969
0
I don't see where is the ambiguity. We keep going around in circles. You ask the same question. We give you the same answer.

After a number of n1 tosses, you get one result, H1 : (n1-H1).

After another number of n2 tosses, you get another result, H2 : (n2-H2).

It does not matter which side H or T has a lead for each experiment.
When you do another trial of 1000,000,000 tosses AND include the results of the previous n1 and n2 tosses, the results H1 and H2 are buried in the noise of the statistical data.
@MrChips,

You say yo do not see the ambiguity

Can I highlight that we say that's its an apparent ambiguity

that if you can not see the question,

then why do yo keep posting the same answer ?

#### Tesla23

Joined May 10, 2009
543
We have some absolute truths with tossing a fair coin where 100 % of all tosses are either H or Tail.

1) The odds at every toss are 50:50 that it will be a tail
2) Each toss is independent of the last, the coin has no memory
3) Over a long enough set, there will be the same number of H as T tossed
4) At any given time, there can be more H than T ( or vice versa )
On the chance my memory is correct:

Point 3 is not true.

If you have an experiment where you make N tosses, the mean number of heads is N/2 and the variance of the number of heads is N/4, so the standard deviation is $$\frac{\sqrt{N}}{2}$$

So, in absolute terms, the more tosses you make, you are most likely to be within a few standard deviations (choose your probability), ±1 standard deviation has width √N. So this gets wider the more tosses you make. The chance of exactly hitting 50:50 gets vanishingly small.

On the other hand, ±1 standard deviation has a relative width of $$\frac{\sqrt{N}}{N} = \frac{1}{\sqrt{N}}$$, which gets smaller as N increases - so you get a better estimate of the fairness of the coin.

#### Deleted member 115935

Joined Dec 31, 1969
0
Thank you @Tesla23

That is a fantastic explanation .

We stated the cases , and yes (3) and (4) seem to be the nub

Your explanation of variance and Standard deviation is abs perfect as far as i can see.

The users sees that over thousands of tosses there are the same number of H as T, WITHIN the tolerance / deviation they are expecting.

That is a great explanation

as yo say, the more tosses the difference effectively becomes vanishingly small,

I did not know about the root N over N ( and I don't know how to type that on the board as well as yo did )

Again , thank you.

#### BobTPH

Joined Jun 5, 2013
9,278
Can I highlight that we say that's its an apparent ambiguity
When you have a false premise, as I pointed out, nothing you deduce is reliable. You are seeing an “ambiguity” because you are are starting from a false premise.

Bob

#### Deleted member 115935

Joined Dec 31, 1969
0
When you have a false premise, as I pointed out, nothing you deduce is reliable. You are seeing an “ambiguity” because you are are starting from a false premise.

Bob
Please see my thoughts on single line replies above,

Once again thank you

#### MrAl

Joined Jun 17, 2014
11,723
this is a though experiment,
please lets not get off track
or upset .

We have some absolute truths with tossing a fair coin where 100 % of all tosses are either H or Tail.

1) The odds at every toss are 50:50 that it will be a tail
2) Each toss is independent of the last, the coin has no memory
3) Over a long enough set, there will be the same number of H as T tossed
4) At any given time, there can be more H than T ( or vice versa )

Which leads to the points in other posts, that there is a calculatable probability that for say 1000 tosses there will be 500 H, or 499 H or 498 H etc.

Given 999 tosses,
say we have 490 H and 509 Tails. We can find the probability of that.
say we have 491 H and 508 Tails. We can find the probability of that.

I now have a conundrum,

Given the 490 H case above,
the next toss is 50:50 a H.

But we also know that over time there should be the same number of H as T

So over the next "n" samples there must be more H than T to move towards the 50:50,
if there were not more H, then T would dominate, and the coin is not fair

But we have said the coin is fair, and it has no memory

How do we square that circle
I think i know where you are going with this, what makes you think this way. There seems to be a 'secondary' law at work here and although i cant explain it i can describe it to some degree.

The first thing si that the 50 percent rule only holds exactly for an infinite number of tosses. For anything less than that there is no rule that can be followed exactly. However, i think most of us here realize that if you see 10 heads in a row it will be more likely that a tails will come up next, even though there is no math that says there will be that i know of. If that doesnt work for you, then what about 100 heads, will the next be a tails? There is something somewhere that must tell us that as the number of samples in a 'run' (as they are called) increases chaos will generate a tails pretty soon.
Now in math there may be nothing that proves this (although there could be) most of us here would have the gut feeling that we are about to get a tails after 100 consecutive heads. I actually did some real life experimenting with this back in the 1980s and found that it is possible to get a very large run of heads or tails, a very unexpectedly large number, but sooner or later the opposite side comes up, every time. In other words, it is very very hard to 'believe' that we would see a run of heads or tails for 1 million tries. So there must be some secondary law at work and i think that would be simply chaos.

To see this happen in real life without having to toss a coin and get continuous 'proof' of a sort that there must be chaos (or something else), build a white noise generator that generates binary codes that are considered random and just use the lowest bit to drive a speaker. That means that the speaker cone jumps out for a '1' and jumps back in for a '0' but because of that it generates sounds with the fundamental frequency of that generated rectangular wave and some odd harmonics for the most part (and some other things not important to this discussion). The question now is simply what do we hear coming from the speaker? We hear white noise, even though we are just using a 1' and a '0' which is probable around 4 volts for a 1 and 0.2 volts for a 0 or close to that, and a series resistor to limit the current into the speaker and out of the IC port.

Now what we can deduce from this is that if there could be a run of a million or millions then we would at some point hear no sound whatsoever because it would generate a string of either 1's or 0's that was too long to create a sound that could be detected by a human, and if it happened often we would just hear 'clicks' not some frequency, even if low. It is true that we may hear very low frequencies but it would always be followed by a tone of some frequency that could be heard by us.

What we can deduce from THAT is if that is not true, that white noise generator would not do us any good for testing anything or just generating a white noise audio generator.

The only catch i can think of is can we actually build a true random white noise audio generator. I know it is possible to build a pseudo random one though. However if you can think of anything else mention it here.

#### boostbuck

Joined Oct 5, 2017
570
i think most of us here realize that if you see 10 heads in a row it will be more likely that a tails will come up next
And there it is again - the gamblers fallacy! As it was expressed in the first post, as an ambiguity. The ambiguity and consequent expectation of an outcome:

"The gambler's fallacy is the belief that the probability for an outcome after a series of outcomes is not the same as the probability for a single outcome. "

#### boostbuck

Joined Oct 5, 2017
570
Or perhaps an alternative way to look at the gambler's fallacy:

You choose a defined random series of H and T ("HHTHTHHTT...") such that the total split is 50/50.
I chose a defined random series of H and T ("HHHHHHHHH.....") such that the total split is 100/0.

You maintain (although you might not see this) that your series is "more likely" than mine. Actually each of the DEFINED series has the same probability.

#### MrAl

Joined Jun 17, 2014
11,723
And there it is again - the gamblers fallacy! As it was expressed in the first post, as an ambiguity. The ambiguity and consequent expectation of an outcome:

"The gambler's fallacy is the belief that the probability for an outcome after a series of outcomes is not the same as the probability for a single outcome. "
Hello,

You obviously dont understand this point of view.
I call that, "The fallacy of the continuous quote of the gamblers fallacy".

Did you ever try this? Go ahead, flip a coin and log the results. You will see long runs come out. But try getting a run of 1000 and see how hard that is to obtain.

It is probably called "The gamplers fallacy" because in any real gambling game you usually dont have enough tries to see a favorable outcome which in this case would simply be seeing the end to a long run. In one procedure, the bet is doubled after each loss so that eventually the gambler wins his original bet and thus makes a little money. But in real life a run can be long enough such that he quickly runs out of money because of the exponential nature of his betting strategy vs the somewhat linear nature of the stream of heads and tails. In other words, there are runs that do occur that are long enough to run anyone out of money before they win but not quite as long i am suggesting here. But you really need to try this in real life in some manner in order to get a feel for what actually happens vs what theory might seem to suggest. It is true that a run of a million can occur, but how often would that really happen. In all of my real life experiments, i have never seen a run so long that i got tired of doing the experiment. I proved that long runs could occur, but not so long that we could start to feel that there was something wrong with the coin.

Another way of looking at this is given a random number generator what are the odds that it will spit out all odd numbers for 24 hours at the rate of 1 new number per 86.4ms.
If we could build a random number generator that could do that, the period of the LSB digit would be 24 hours, which would mean a frequency of about 11.6 microhertz. What's more intriguing is what are the odds that it will do it again. In theory it should be possible, but seriously it is hard to believe it will happen over and over again for years to come in REAL life.

I think this is related to pattern generation through a random process. What are the odds that a four digit lottery number will come out to the same number twice in a row. I've looked back some months and cant find any, but in theory it should happen. So what if it does, then what are the odds that it will come out a third time, a fourth time, etc. In theory it should but try finding that kind of run.
What i have seen is four digits come out all the same number, like 9999. That should happen once every 10000 games, but why so rare.

#### Deleted member 115935

Joined Dec 31, 1969
0
The conclusion I make , is there are various "laws" at work

probability say that a run of 100 H is very unlikely
Chance also say that each throw is independent ( no memory ) so each throw is 50:50 a H

The extra, was the "rule" that over time the number of H and T is equal,
as was pointed out, this is true BUT within the limits of resolution
i.e if you toss twice, then its 100 % true that you will have the same number of H as Tail, +-1

But when you toss 1Million times, the number of H and T is very unlikely to be exactly 500K,
but within say +-100 its very likely that they numbers are the same,

( some one can do the maths, but that's good enough for my old brain )

#### BobTPH

Joined Jun 5, 2013
9,278
The extra, was the "rule" that over time the number of H and T is equal,
as was pointed out, this is true BUT within the limits of resolution
I am sorry, but, for the fourth time, your “law” is provably false. We can calculate, via the probability theory, that the larger the number of the lower the probability that the H an T are equal.

And Mr Al is equally wrong in arguing that it converges at infinity. A series that starts at 1/2 and is monotonically decreasing cannot converge to 1! In fact, for an infinite number of trials, the probability of an equal number of H and T is infinitesimal.

Bob

#### bogosort

Joined Sep 24, 2011
696
You obviously dont understand this point of view.
I call that, "The fallacy of the continuous quote of the gamblers fallacy".
And you obviously don't understand that coin flips are independent events. You literally said " i think most of us here realize that if you see 10 heads in a row it will be more likely that a tails will come up next", which is not only wrong, it's precisely the gambler's fallacy.

Did you ever try this? Go ahead, flip a coin and log the results.
I've noticed that people who suggest actually flipping coins instead of "trusting" the math of probability implictly believe that something else is going on with physical coin flips that the math doesn't capture. It will save everyone time if you just state directly what you think is missing from the mathematical analysis.

You will see long runs come out. But try getting a run of 1000 and see how hard that is to obtain.
The probability of seeing a run of 1000 heads can be made as close to 100% as you wish simply by increasing the number of trials. But this is true of any particular sequence, and so a run of 1000 heads is no "harder" than any other 1000-flip sequence. Your mistake is in assuming that, after having seen 999 consecutive heads in n flips, the (n+1)th flip is more likely to be T than H.

I propose a logical (rather than mathematical) argument against the gambler's fallacy. Since 1000 heads in a row is considered "hard", then 999 heads in a row must also be hard. Therefore, after seeing 998 heads in a row, the next flip is more likely to be tails. But if 999 heads in a row is hard, then so must be 998 heads in a row. Therefore, after seeing 997 heads in a row, the next flip is more likely to be tails.

After seeing 996 heads in a row, the next flip is more likely to be tails.
After seeing 995 heads in a row, the next flip is more likely to be tails.
...
After seeing 3 heads in a row, the next flip is more likely to be tails.
After seeing 2 heads in a row, the next flip is more likely to be tails.
After seeing 1 head, the next flip is more likely to be tails.

Because nothing about the coin changes between a run of 998 or a run of 1, belief in the first statement implies belief in the last statement. But since the last statement is obviously false, the first must also be false. QED.

#### Deleted member 115935

Joined Dec 31, 1969
0
I am sorry, but, for the fourth time, your “law” is provably false. We can calculate, via the probability theory, that the larger the number of the lower the probability that the H an T are equal.

And Mr Al is equally wrong in arguing that it converges at infinity. A series that starts at 1/2 and is monotonically decreasing cannot converge to 1! In fact, for an infinite number of trials, the probability of an equal number of H and T is infinitesimal.

Bob
@BobTPH

I see they are not one line any more, which I think is great

I could get into the meaning of "law" and theory
especially with one in quotes, i.e implicitly recognising its not a law,

And I'm certain you would come up with some more great one line responses,

I think you are saying the same thing again and again , so not adding to the conversation
which by definition is not adding to the conversation

especially when you start with the "assertive" wording " but, for the fourth time"

There is an old saying about English abroad I heard from an old friend

something on the lines of

'They do not listen to the locals who don't speak their language,
, and just shout louder and louder the same thing'

#### BobTPH

Joined Jun 5, 2013
9,278
@BobTPH

I see they are not one line any more, which I think is great

I could get into the meaning of "law" and theory
especially with one in quotes, i.e implicitly recognising its not a law,

And I'm certain you would come up with some more great one line responses,

I think you are saying the same thing again and again , so not adding to the conversation
which by definition is not adding to the conversation

especially when you start with the "assertive" wording " but, for the fourth time"

There is an old saying about English abroad I heard from an old friend

something on the lines of

'They do not listen to the locals who don't speak their language,
, and just shout louder and louder the same thing'
I think you are refusing to believe what everyone else in telling you and continuing to go with your "intuition" despite mathematical and logical proofs to to the opposite.

I agree, we both keep repeating the same thing. The difference is that what you keep repeating is false, while what I keep repeating is true.

And I will repeat once again, that starting with a false premise is what it causing all your confusion.

Bob

Status
Not open for further replies.