understanding probability

[Eric]

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.

> 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.
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