Comment author:sharpneli
13 December 2009 12:31:54AM
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Yvain said the finiteness well, but I think the "infinitely many possible arrangements" needs a little elaboration.

In any continuous probability distributions we have infinitely many (actually uncountably infinitely many) possibilities, and this makes the probability of any single outcome 0. Which is the reason why, in the case of continuous distributions, we talk about probability of the outcome being on a certain interval (a collection of infinitely many arrangements).

So instead of counting the individual arrangements we calculate integrals over some set of arrangements. Infinitely many arrangements is no hindrance to applying probability theory. Actually if we can assume continuous distribution it makes some things much easier.

Comment author:sharpneli
13 December 2009 10:40:09AM
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It does work, actually if we're using Integers (there are as many integers as Rationals so we don't need to care about the latter set) we get the good old discrete probability distribution where we either have finite number of possibilities or at most countable infinity of possibilities, e.g set of all Integers.

Real numbers are strictly larger set than integers, so in continuous distribution we have in a sense more possibilities than countably infinite discrete distribution.

## Comments (78)

BestYvain said the finiteness well, but I think the "infinitely many possible arrangements" needs a little elaboration.

In any continuous probability distributions we have infinitely many (actually uncountably infinitely many) possibilities, and this makes the probability of any single outcome 0. Which is the reason why, in the case of continuous distributions, we talk about probability of the outcome being on a certain interval (a collection of infinitely many arrangements).

So instead of counting the individual arrangements we calculate integrals over some set of arrangements. Infinitely many arrangements is no hindrance to applying probability theory. Actually if we can assume continuous distribution it makes some things much easier.

Good point. Does this work over all infinite sets, though? Integers? Rationals?

It does work, actually if we're using Integers (there are as many integers as Rationals so we don't need to care about the latter set) we get the good old discrete probability distribution where we either have finite number of possibilities or at most countable infinity of possibilities, e.g set of all Integers.

Real numbers are strictly larger set than integers, so in continuous distribution we have in a sense more possibilities than countably infinite discrete distribution.