heads or tails ........

Status
Not open for further replies.

Thread Starter

andrewmm

Joined Feb 25, 2011
1,456
So can I start with
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
 

BobTPH

Joined Jun 5, 2013
3,304
Read and understand rule 2 in your original post.
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
Can this be true, given rule 2?

Bob
 

jpanhalt

Joined Jan 18, 2008
11,088
The probability any specific distribution in a situation with many possible states is quite low. Some sites devoted to those calculations have been presented here before.
 

MrChips

Joined Oct 2, 2009
23,503
There is no conundrum.

50:50 H or T is a probability, not a rule.

My AC LINE frequency gives me 59.5Hz in the first 12 hours owing to uncertainty.
In the next 12 hours I get 60.5Hz. So I still get 60Hz in 24 hours. That is a rule.
 

Thread Starter

andrewmm

Joined Feb 25, 2011
1,456
Read and understand rule 2 in your original post.


Can this be true, given rule 2?

Bob

Thank you @BobTPH,
Can I remind you that I said at the front that these are rules, I understand them and do not argue with them,

You are saying that ever toss is a 50 % chance of being a H,
How does that square with the statement that over a sufficient run,
the number of H and T will be the same,
 

Thread Starter

andrewmm

Joined Feb 25, 2011
1,456
There is no conundrum.

50:50 H or T is a probability, not a rule.

My AC LINE frequency gives me 59.5Hz in the first 12 hours owing to uncertainty.
In the next 12 hours I get 60.5Hz. So I still get 60Hz in 24 hours. That is a rule.
An interesting point what is a rule, a law ( as in ohms law ) ,
is a probability a manifestation of a rule / law,

Re the AC line frequency, that hardly meets the test for randomness.

But anyway I don't think it addresses the question,
 

MrChips

Joined Oct 2, 2009
23,503
Probability and Statistics
Statistics of small samples vs large samples

Statistics obey probability not rules.

Ohm's Law is the manifestation of a probability.
It becomes a rule or law when applied to very large numbers, e.g. 1A = 6.242 x 10^18 electrons per second.

Flip 6 x 10^18 coins and you start to see the picture.
 

djsfantasi

Joined Apr 11, 2010
7,689
So over the next "n" samples there must be more H than T to move towards the 50:50,
Your “circle” is already a “square”.

That is, your statement is false. So, it can’t be used to make a follow on conclusion. By your rules, the coin is fair and the only conclusion that you can draw us that there is a 50:50 chance of H vs T on every toss. You said “there must be...” But since there is no memory, there can’t be, since each toss knows not of the previous toss results.
 

Thread Starter

andrewmm

Joined Feb 25, 2011
1,456
@djsfantasi

Im in totaly agreement the coin is fair,
and each toss if 50:50

Im also of the idea that over a reasonable time the result will average to the same number of H as T

so for 999 tosses if were not at 50:50
then by 100000 tosses we should be closer to 50:50


Hence the question, if were at 999 tosses, and have already 490 H,
to get nearer to 50:50 by 100000 tosses, must there be more H than T ?
 

MrChips

Joined Oct 2, 2009
23,503
Here is another explanation.

Do 1000 flips. Record the number of H and T.
Do another 1000 flips. The result is independent on the previous trial. There is no memory. There is no catching up to do.
The outcome is based on the probability of 1000 independent flips.

Repeat this 1,000,000 times, i.e. a total of 1000 x 1,000,000 flips.
Each trial is independent of each other trial. Yet you will get closer to your expectation of 50:50.
 

MrChips

Joined Oct 2, 2009
23,503
@djsfantasi
Hence the question, if were at 999 tosses, and have already 490 H,
to get nearer to 50:50 by 100000 tosses, must there be more H than T ?
No. Each trial is independent of any previous trial. There is no memory.
The probability is still 0.5 for each independent flip.
You are confusing probability with statistics.
 

wayneh

Joined Sep 9, 2010
17,152
Hence the question, if were at 999 tosses, and have already 490 H,
to get nearer to 50:50 by 100000 tosses, must there be more H than T ?
No. In fact you'd have to predict going forward that your discrepancy will never go away. It will only be diluted. An anomaly in the past does not affect the future.

Regression toward the mean is a real thing, but it does not require a coin to have a memory.
 

Thread Starter

andrewmm

Joined Feb 25, 2011
1,456
No. In fact you'd have to predict going forward that your discrepancy will never go away. It will only be diluted. An anomaly in the past does not affect the future.

Regression toward the mean is a real thing, but it does not require a coin to have a memory.
I agree,
As said at the very beginning,
(2) the coins have no memory

Do we agree, that over along enough run of tosses the number of H and T will be "equal" to within some margin ?
 

MrChips

Joined Oct 2, 2009
23,503
Here is another example of why your logic is flawed.

Flip a coin 100 times.
The probability of getting 40 heads is 0.01 or 1%.

By your argument there ought to be a higher probability of getting 60 heads in the next 100 flips.
The probability of getting 60 heads in 100 flips is still 0.01 or 1%.
 

Thread Starter

andrewmm

Joined Feb 25, 2011
1,456
No. Each trial is independent of any previous trial. There is no memory.
The probability is still 0.5 for each independent flip.
You are confusing probability with statistics.
Totaly agree ,
as said in the original post
(2) each toss is independent , 50:50 probability.

Might be off topic, but whats the definition of probability and statistics ?
 

wayneh

Joined Sep 9, 2010
17,152
I agree,
As said at the very beginning,
(2) the coins have no memory

Do we agree, that over along enough run of tosses the number of H and T will be "equal" to within some margin ?
Only by dilution, ie. by percentage. If you're 10 heads over at some point, there is no reason to think that lead will ever disappear.
 

MrChips

Joined Oct 2, 2009
23,503
I agree,
As said at the very beginning,
(2) the coins have no memory

Do we agree, that over along enough run of tosses the number of H and T will be "equal" to within some margin ?
In the long run, yes.
Each short run is still independent of each other short run. There is no memory.
You need to look at the statistics of large samples.
 

Thread Starter

andrewmm

Joined Feb 25, 2011
1,456
Only by dilution, ie. by percentage. If you're 10 heads over at some point, there is no reason to think that lead will ever disappear.
Good point,

at 20 toss, if H is 10 ahead, that's a lot, at 100000 toss 10 ahead its nothing,

Wonder if thats a clue ?
 
Status
Not open for further replies.
Top