understanding probability

[Christopher Browne]

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.
>
> 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.

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.

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.

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.

Practical cryptography is troublesome, because you tend to have
a limited set of messages to work with, so it's rather difficult to
infer between things like that a), b) and c).

Naive users of cryptography (and I think that pretty much includes
all of us here; we haven't had to use crypto to protect messages
against life-and-death-important attacks) don't know what all
things to try to guard against.

>> 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.

I imagine that casinos have an active desire to prefer shuffling
algorithms that tend to leave cards in "disarray" such that they
expect the patterns involving lots of sequences to be particularly
unlikely.

After all, if players notice that, after a shuffle, the cards are
coming out in what they *imagine* to be an interesting order (and I
love your Feynman quote here!), someone's going to get beat up.

Truly being random isn't as good, in this case, as *appearing* random.
-- 
When confronted by a difficult problem, solve it by reducing it to the
question, "How would the Lone Ranger handle this?"
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