# Math Help - Coin tossing and bounds

1. ## Coin tossing and bounds

Here is my conjecture.
If we toss a fair coin n times (with n>=100) then the longest run of heads (or tails) will always be < or = to int(sqrt(2n)).
Example :
n=100
Let r be the longest run of heads or tails
r<=int(sqrt(200))
r<=14
Please try to simulate I need a feedback. The formula is easy to compute but does it fit what a long calculation can do.
Thank you for any proof or clue on how to prove it.

2. ## Re: Coin tossing and bounds

That is certainly not true. If you flip a fair coin n times, then the "longest run of heads or tails" can be n. That has a low probability ( $\frac{1}{2^n}$) but it can happen.

3. ## Re: Coin tossing and bounds

Originally Posted by HallsofIvy
That is certainly not true. If you flip a fair coin n times, then the "longest run of heads or tails" can be n. That has a low probability ( $\frac{1}{2^n}$) but it can happen.
What is the probability that my conjecture is not true? 100%? 99%? o%?
What is sure (100%) is that my conjecture is not 100% false.

4. ## Re: Coin tossing and bounds

If you toss 10.000 times a coin and the outcome is 10.000 heads what about fairness?
How do you test fairness of any coin?
It can happen too that the combination of 5 Euromillions numbers drawn twice a week will repeat itself 10000000000000000000000000000000000000 times.
It can happen too.
So what is the purpose of the probability theory if any event can happen no matter how low is its probability?
Try to simulate using either real coin or any random function.

5. ## Re: Coin tossing and bounds

Mouhaha
Last week one american won 350 million dollars? in lotto....this is luck but such things happens very rarely...as Hall noticed it can be n times the same result but the probability is there.....it does not mean that it will happend very soon again.....but it happens....the lucky american was playing the same number for more than 15 years!!!!...

6. ## Re: Coin tossing and bounds

I ran the test for 10.000 times during hours. The longest run of tails was 36 so far from int(sqrt(20000)=141.

7. ## Re: Coin tossing and bounds

So what is the purpose of the probability theory if any event can happen no matter how low is its probability?
Events with probability 0 cant happen.

the "purpose" of probability theory is to provide a framework to understand events which are uncertain. This is useful for, among other things, weather forecasting, finance, insurance, and many other things.

How do you test fairness of any coin?
There are well established methods for doing this (Checking whether a coin is fair - Wikipedia, the free encyclopedia) but you should understand the basics of probability theory before looking at that.

I ran the test for 10.000 times during hours. The longest run of tails was 36 so far from int(sqrt(20000)=141.
what do you deduce from this?

8. ## Re: Coin tossing and bounds

Originally Posted by SpringFan25
Events with probability 0 cant happen.

the "purpose" of probability theory is to provide a framework to understand events which are uncertain. This is useful for, among other things, weather forecasting, finance, insurance, and many other things.

There are well established methods for doing this (Checking whether a coin is fair - Wikipedia, the free encyclopedia) but you should understand the basics of probability theory before looking at that.
What if I say that the basics of probability are completely wrong?

9. ## Re: Coin tossing and bounds

What if I said that the basics of probability are completely wrong?
Like any other academic field you are welcome to prove such a statement, however nothing in this thread does so.

Why exactly do you think it is wrong? what evidence do you have to support that?

10. ## Re: Coin tossing and bounds

You can build any system (in any fields ) on the basis that there 1 chance out of 2^999999999999999999999999999999999999999999 that it fails and surprisingly you just started to use you system big crash!!!

11. ## Re: Coin tossing and bounds

if you think that probability theory predicts that "extremely unlikely events never happen" than you have not understood the theory.

if things with low probability happen rarely in independant, randomized trials then the theory is predicting the pattern of events correctly.

I honestly cant tell if you are joking or not, but supposing your example is serious then in the "real world" it would go something like this:

• An engineer assembles the machine.
• he does a study and concludes it has a 99.99999% chance of working
• You turn it on and it breaks

What do you deduce from this? Does this prove that "probability theory is wrong"? or does it just suggest the probability was incorrectly estimated at 99.99% in the first place?

12. ## Re: Coin tossing and bounds

Originally Posted by SpringFan25
Like any other academic field you are welcome to prove such a statement, however nothing in this thread does so.

Why exactly do you think it is wrong? what evidence do you have to support that?
Sorry! I will come back (busy for now).

13. ## Re: Coin tossing and bounds

I got a phone.
Let me put it my way.
Any logical system built on axioms, hypothesis, conjectures and so on is not wrong in itself.
It is tautological.
An equation is an identity. Even when you seems to find a solution you did not find nothing. You are just another way what really is "in".
As we say in french : "tu ne peux y trouver que ce tu y apportes comme lauberge espagnole"
You use the infinity concept, the indepedency of events and so on as basis of thinking.
But do not forget that infinity, independency are not "real".
The roulette, the coins and other systems are physical machines producing something we call "random", unpredictable.
What make me think that the probability theory is wrong is the fact that we use it in real life.

14. ## Re: Coin tossing and bounds

Originally Posted by Mouhaha
I got a phone.
Let me put it my way.
Any logical system built on axioms, hypothesis, conjectures and so on is not wrong in itself.
It is tautological.
I'm not quite sure what you are trying to say here but it has been shown, by Kurt Goedel, that any system of axioms, large enough to include the definition of the natural numbers, is either inconsistent or incomplete.

An equation is an identity. Even when you seems to find a solution you did not find nothing. You are just another way what really is "in".
As we say in french : "tu ne peux y trouver que ce tu y apportes comme lauberge espagnole"
You use the infinity concept, the indepedency of events and so on as basis of thinking.
But do not forget that infinity, independency are not "real".
The roulette, the coins and other systems are physical machines producing something we call "random", unpredictable.
What make me think that the probability theory is wrong is the fact that we use it in real life.
You believe it is wrong because it is used? That seems very strange! We use addition in real life also. Does it follow that addition is wrong also?

15. ## Re: Coin tossing and bounds

Originally Posted by SpringFan25
if you think that probability theory predicts that "extremely unlikely events never happen" than you have not understood the theory.

if things with low probability happen rarely in independant, randomized trials then the theory is predicting the pattern of events correctly.

I honestly cant tell if you are joking or not, but supposing your example is serious then in the "real world" it would go something like this:

• An engineer assembles the machine.
• he does a study and concludes it has a 99.99999% chance of working
• You turn it on and it breaks

What do you deduce from this? Does this prove that "probability theory is wrong"? or does it just suggest the probability was incorrectly estimated at 99.99% in the first place?
This can prove both.
Why are they hiding something maybe deeply wrong behind some miscalculation?
If the probability theory can not predict unlikely events then what is its utility.
Sometimes I question myself : why the probabilists are making a distinction between tossing 10 times a coin and tossing a coin 50000000000000000000 times?
Any toss is identical to another one : is a permuted one.
You replace in lotto numbers 1 by 6 or 9 by 12 they have to be identical. Imagine that you paint the balls with another value and you pick it : 9 could 15, 10 could be 1...
So the same combination is drawn every time, the only difference is that we assign to the balls a value before picking them. If you permute it in your mind then any ticket sold is the winning one.

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