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#### MrAl

Joined Jun 17, 2014
11,693
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
Hi,

Not sure what you are saying with your first statement.
It is interesting though that with your second statement starting with "And MrAl..." sounds like it is just the opposite of what i am saying. I am saying that if you generate random numbers 0 through 10 and add up the numbers and divide by the number of times you generated these samples that the average would get closer and closer to 5.
So for heads and tails calling heads 1 and tails 0, that would mean that if you tossed a lot of times the more you tossed the closer the sum divided by the number of times would be 1/2. So i dont know what you mean by 'converting to 1'. or whatever you are saying there. What that means is that for a very large number of tosses the number of heads would equal the number of tails very closely, and it seems logical that as you toss more and more each single contribution becomes less and less important, much like a low pass filter with a very long time constant, and so once we get near the 1/2 way point it would start to fluctuate less and less from that mean.

This is another case of "try it and see". On the computer i tis easy to generate pseudo random numbers, sum the results of N samples, and divided by N. The result will be the mean of the allowed random numbers (like for integers 0 through 10 the mean is 5).
To state that it would get 'farther' from the mean does not make any sense to me, but i cant be sure what you are saying because you seem to think that something will somehow converge to '1' when i dont think i said that or implied that.

I could easily post some experimental results to show how this works.
A really quick example would be the first 17 digits of pi:
3.1415926535897932
Add them up and divide by 17 and we get 4.7647 approximate. The mean would be 4.5 which is close already.

#### ericgibbs

Joined Jan 29, 2010
19,091
hi,
This is typical of guessing how many jelly beans in a jar.
Averaging the answers given by a number of other 'guessers' can result in a winning guess,

E

#### BobTPH

Joined Jun 5, 2013
9,263
The first thing si that the 50 percent rule only holds exactly for an infinite number of tosses.
This is tje statement that I am disagreeing with. The 50% “rule”, as stated by the TS never holds true. There is no rule that the number of H will eventually be equal to the number of T. And, as I have shown, the probability of an equak number ig H and T goes down with more trials.

As for your average. At 2 trials the average number of H and T over all outcomes us already exactly equal, and it remains so for all so for any even number of trials, so saying it is equal for infinity adds no new information.

Bob

#### Deleted member 115935

Joined Dec 31, 1969
0
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
@BobTPH

I don't know who that one line attack was aimed at

I hope not me

It seems that you have miss understood the original question I posed

I have asked for this post to be locked as the comments from people such as yourself are being as you yourself say is repetitive and as such not adding anything to the conversation

Thank you to all those that have understood the original question

Which is about trying to find a possible reason why the original statements can all be true but at first sight appear to come up with a contradiction.

And thank you @BobTPH , the reason for the contradiction is not because all those that come to it a "thick" or are "gamblers"

I hop that this is the end of this ..

#### Deleted member 115935

Joined Dec 31, 1969
0
This is tje statement that I am disagreeing with. The 50% “rule”, as stated by the TS never holds true. There is no rule that the number of H will eventually be equal to the number of T. And, as I have shown, the probability of an equak number ig H and T goes down with more trials.

As for your average. At 2 trials the average number of H and T over all outcomes us already exactly equal, and it remains so for all so for any even number of trials, so saying it is equal for infinity adds no new information.

Bob
@BobTPH,
i think we have agreed that there will be a the same number of H as T , given an infinite number of tosses,
and at any point, the probability of not being 50:50 is a number, which gets smaller the further away from 50:50 you are

For 1000 tosses, its more likely to have between 450 and 550 H than it is to have 999 H and one T

some one can work out the probability of each of these cases.

Thank you as ever @BobTPH , I di think that this puts an end to my thread
May be if you have other thoughts you could start your own discussion.

#### bogosort

Joined Sep 24, 2011
696
It's a consequence of the central limit theorem.
If you're agreeing with Bob that there is no such law (i.e., |H| = |T|) because of the CLT, then good point.

The probability that |H| = |T| approaches 0 as n → ∞, i.e., the more you toss a coin, the less likely you will have equal heads and tails. This is indisputable. Those who claim that "something" will eventually cause the number of heads to equal the number of tails need to explicitly state what that "something" is.

#### bogosort

Joined Sep 24, 2011
696
i think we have agreed that there will be a the same number of H as T , given an infinite number of tosses,
No one should agree with this statement!

As a side note, it helps to be precise with language, especially when reasoning about such topics. There is no number of tosses that equals ∞, so it isn't meaningful to claim that |H| = |T| after ∞ tosses. Rather, we should say that as the number of tosses increases without bounds ("approaches infinity"), the probability that the number of heads (|H|) equals the number of tails (|T|) approaches zero.

#### 402DF855

Joined Feb 9, 2013
271
It's fairly easy to prove that the number of heads equals the number of tails as the number of samples approaches infinity. If tails and heads are denoted as 0 and 1 then as the number of flips approaches infinity then the mean will be equal EXACTLY 0.5 (by the central limit theorem). This proves that number of heads equals number of tails. The fact that both are infinite is not important. Infinities can be equal or not equal.

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