Re: "Happens with Paobability 1"

On Aug 23, 4:57 am, "S.W.Christensen" <s...@xxxxxxxxxxxxxxxx> wrote:
On 22 Aug., 20:10, Randy Poe <poespam-t...@xxxxxxxxx> wrote:

Technically, probability 1 guarantees "almost certainty"
because there may be other allowed outcomes (with probability
0).

On a continuous distribution, every particular outcome
has zero probability.

I disagree with this, but first a disclaimer: I'm not a pure
mathematician, and I don't work on the foundations. My area is applied
mathematics (including applied Bayesian probability theory).

It seems to me that it is meaningless to say that the probability of
any particular event is zero, if the event is possible. Surely zero
must simply be a limit for the distribution, and therefore
unreachable.

This raises the question: Is this disagreement simply a consequence of
the use of infinite sets, versus the non-use of infinite sets?

In a word, yes.

Since real world measurements have uncertainty, I could
easily see a viewpoint where the only possible outcomes
of a uniform random number generator were intervals
encompassing measurement uncertainty. And the probability
of a number being in any particular interval of width
a>0 is a.


If it
is, then maybe my critique is misdirected, because you specifically
mentioned a "continuous" distribution, which I reject because I
consider continuity ungraspable.

Well then, you're throwing out a great deal of useful
analytical tools, including calculus.

- Randy

.

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