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Infinite Flips


Best Answer bonanova, 11 December 2013 - 06:40 PM

Spoiler for Suggestive proof

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#11 bonanova

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Posted 09 December 2013 - 08:46 PM

Once bias exists, infinity won't guarantee a return to equality.

The probability of equality decreases with bias, but it's never zero unless one of the outcomes never occurs at all.

Note that saying p(H=T) < 1 doesn't prohibit H=T, it only says H=T is not a certain eventual outcome.


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#12 Rainman

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Posted 09 December 2013 - 09:40 PM

Just a side note: it is often falsely assumed that 100% probability is equivalent with absolute certainty. This is one of those cases; even if the coin is perfectly fair, it's possible that you will never reach H=T even if you keep flipping forever. The probability is 0, but it's possible. Any given infinite sequence is possible, for example flipping heads every time.

 

If we look at it backwards, suppose you did flip the coin infinitely many times and recorded your results. Whatever sequence you got, the probability of that particular sequence was 0, but it did happen.

 

Suppose you generate a random real number between 0 and 10. The probability of generating pi is 0, but it is within the realm of possibilities. It's totally impossible to generate -pi.

 

In general, suppose we have an event A. If A is absolutely certain to happen, then P(A)=1. But the converse is not true. If P(A)=1, we can't conclude that A is absolutely certain to happen. There is a definition in probability theory, that if P(A)=1, then A happens almost surely.


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#13 plainglazed

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Posted 10 December 2013 - 01:54 AM

well, i almost surely better understand now. it's actually a little easier for my wee brain to comprehend.  the learning experience here is always worth a little humility imho.  thanks you guys.  oh, one last (not serious) question - what if infinity is odd? 


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#14 bonanova

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Posted 10 December 2013 - 04:46 AM

well, i almost surely better understand now. it's actually a little easier for my wee brain to comprehend.  the learning experience here is always worth a little humility imho.  thanks you guys.  oh, one last (not serious) question - what if infinity is odd? 

 

Nice segue to a cute poem from the late Martin Gardner:

 

Pi goes on and on and on ...

And e is just as cursed.

I wonder: Which is larger

When their digits are reversed?


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#15 bonanova

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Posted 10 December 2013 - 04:54 AM

Spoiler for My best shot at a proof

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#16 bonanova

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Posted 11 December 2013 - 06:40 PM   Best Answer

Spoiler for Suggestive proof


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#17 gavinksong

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Posted 12 December 2013 - 12:20 AM

Spoiler for Suggestive proof

 

Woah.


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#18 plasmid

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Posted 12 December 2013 - 04:16 AM

How do you partition the flips of a coin with p(tails)=0.4 and p(heads)=0.6 in the scenario where it comes up with ten consecutive tails?
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#19 bonanova

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Posted 13 December 2013 - 12:52 PM

How do you partition the flips of a coin with p(tails)=0.4 and p(heads)=0.6 in the scenario where it comes up with ten consecutive tails?


T T T T H H H H - H H is a result that is representative of p(H) = 0.6.

That does not mean that in every ten flips it will happen.
There can be (arbitrarily long) strings of T, so long as p(T) is not zero.

So T T T T T T T T T T can happen.
But p(H) = 0.6 says that, on average, whenever there are 10-T strings there will also be 10-H strings, and they will occur in the following proportions:

T T T T T T T T T T
T T T T T T T T T T
T T T T T T T T T T
T T T T T T T T T T
H H H H H H H H H H
H H H H H H H H H H
H H H H H H H H H H
H H H H H H H H H H

-------------------

H H H H H H H H H H
H H H H H H H H H H


and the hyphens indicate how to partition them.
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#20 Rainman

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Posted 13 December 2013 - 10:20 PM

If P(H)=0.6, the probability of a 10-H string is 0.610 ~ 0.006, whereas the probability of a 10-T string is 0.410 ~ 0.0001. So 10-H strings should be much more frequent than 10-T strings.


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