understanding probability

[Eric B]

> ________________________________
>> From: Christopher Browne <cbbrowne-Re5JQEeQqe8AvxtiuMwx3w at public.gmane.org>
>>To: TLUG Mailing List <tlug-lxSQFCZeNF4 at public.gmane.org>
>>Sent: Wednesday, August 7, 2013 9:11:00 AM
>>Subject: Re: [TLUG]: understanding probability
>>
>>
>>On Tue, Aug 6, 2013 at 2:52 AM, Eric <gyre-Ja3L+HSX0kI at public.gmane.org> wrote:
>>> On Sat, 3 Aug 2013, James Knott wrote:
>>>
>>>> However, my understanding about probabilities says that while
>>>> something
>>>> may be unlikely, it's not impossible and, according to chance may even
>>>> happen immediately.
>>>
>>> That is the intuitive paradox.
>>> One could argue that anything that appears to be impossible
>>> is only very improbable, since given a few tweaks you can
>>> make the impossible into something extremely improbable.
>>> However, rational people shouldn't fret about impossible events.
>>>
>>> For example, according to radioactive statistics, it is
>>> possible that all the atoms in any non-supercritical fissile
>>> mass may spontaneous decay, all at once, since each atom has a
>>> finite probability of decaying in the next second.
>>> The total spontaneous decay probability is unimaginably
>>> small.  No scientist is concerned with that because, although
>>> mathematically possible, it is practically impossible.
>>>
>
> Well, my take on infinity (and infinitesimal) is that it is just
> convenient tools to simplify mathematical derivation, otherwise, as a
> concept "the machinery of logical and mathematical reasoning also breaks
> down when applied to infinity"
> [http://www.thelawofphysics.com/table-of-contents/infinity]. That means,
> infinity shouldn't be used in describing reality.

To put it more simply, infinity doesn't exist.

>>> In a finite observable universe, some things will never happen.
>>> And to make the odds even more improbable, the actual realm
>>> where we exist is a tiny tiny fraction of that observable
>>> universe.
>>
>
> But, even without infinity, we know that the universe came about from
> nothing. If the universe and everything in it can come from nothing, what
> can be more surprising?

No one knows that.

>>My concern about the hashing systems is that, in the absence of
>>sufficient analysis, the whole "super-hyper-infinitely-improbable"
>>measures are estimations, and are not certain to be the actuality.

I share your concern that this is the weakness, but I wouldn't say there
has been insufficient analysis.

>>And note that in practice, it's eminently difficult to perfectly validate
>>the precise characteristics of the fissile mass, as what you can
>>know about it tends to come out of statistical analysis of observations.
>>(If we knew when atoms were going to decay, well, that's supposed
>>to be the thing we are completely unable to know, isn't it?!?)
>>
>>If I saw too many atoms decaying all at once, I would be keen to
>>head to the hypothesis that we got the model wrong, and imagined
>>that the material was composed rather differently than was truly
>>the case.

The last thing you should consider is that the physics model is wrong.

>>Similarly, if I "shuffled" a deck, and then drew 4 aces in a row, I'd
>>be pretty inclined to the hypothesis that this wasn't just random
>>chance, but rather:
>>a) Some exploit in the shuffling system, or
>>b) Perhaps there's more than 4 aces in the deck?
>>
>>And if I saw a bunch of SHA-1 collisions, then either:
>>a) Lotta duplicate things getting hashed,
>>b) Buggy implementation of SHA-1, or
>>c) Something more deeply wrong with SHA-1
>>are all plausible hypotheses, and it may be quite difficult to
>>distinguish between b) and c), in particular.

Again, option c) is he least plausible since no one has found a false
collision.

>>>> The astrophysicist Brian Greene has some
>>>> interesting comments on this sort of thing, in the realm of infinite
>>>> universes.  One thing he mentions is that if the universe is truly
>>>> infinite, then some subset of it, such as the portion within our
>>>> observable limits, must be repeated an infinite number of times.
>>>
>>> Yes, within the framework of theoretical infinities,
>>> everything will happen sometime; so what?  That is pure speculation.
>>> That doesn't change the odds within the relatively small
>>> finite (not infinite) realm where we exist.
>>> (Green could be speculating to explain how the laws of
>>> physics appear to be fine-tuned.)
>>>
>>>> Another thing he mentioned was if you shuffle a deck of cards, while
>>>> highly unlikely, one possible outcome is all the cards will be
>>>> properly
>>>> sorted.  This is because no matter how many there are, there are a
>>>> finite number of possible combinations.  Once you've shuffled past
>>>> that
>>>> number of times, and likely well before it, then repeat sequences are
>>>> inevitable.
>>>
>>> In an ideal shuffle, every permutation is equally likely.
>>> There is nothing special about all the cards being properly sorted.
>>> It only seems remarkable because it is easy to recognize.
>>> Or as Feynman satirically pointed out:
>>>
>>>   "You know, the most amazing thing happened to me tonight. I
>>>   was coming here, on the way to the lecture, and I came in
>>>   through the parking lot. And you won't believe what
>>>   happened. I saw a car with the license plate ARW 357. Can
>>>   you imagine? Of all the millions of license plates in the
>>>   state, what was the chance that I would see that particular
>>>   one tonight? Amazing!"
>>>
>>> The fallacy in the card deck premise is the phrase "once
>>> you've shuffled past that number of times".
>>> Since you will never reach that number (52!), it's a
>>> non-problem.
>>
>>In principle, there may be 52! combinations; to be sure, usual
>>shufflings won't get anywhere close to that.

Yes. I was talking about "ideal" shuffling.


--
The Toronto Linux Users Group.      Meetings: http://gtalug.org/
TLUG requests: Linux topics, No HTML, wrap text below 80 columns
How to UNSUBSCRIBE: http://gtalug.org/wiki/Mailing_lists