Infinity and probability

1. Jul 9, 2007

blackbird3

I'd really appreciate it if someone could help me with the point below! It relates to a real philosophical problem but I'm baffled by the maths.

Assuming no other variables apply, if there is infinite space in which a substance *could* exist, let's call it x, and there are not limits to how big or small x is, or whether there is more than one, but there are infinite possibilities of what else could be in the space the x would take up instead of x, including nothing, what are the odds of x existing within the infinite space? By this I don't mean a specific space, I mean what are the odds that there will be x *somewhere* within the infinite space.

(By substance, I really mean thing that exists, not substance made up of atoms and molecules, so without getting into the debate of whether or not colours are substances, it might be helpful to think of x as a tiny spot of red, imagining that spots of red of any given size *could* exist in the space but there are infinite other things that could also exist, and the spaces could also be empty.)

I've tried to work it out different ways and I've come up with infinity to 1 against it being there, infinity to 1 in favour of it being there, it's both simaltaneously infinitely likely and infinitely unlikely, and various other answers... so I'm definitely doing something wrong!

Ask me to clarify if this reads like mush! Vastly appreciate any help!

2. Jul 9, 2007

cristo

Staff Emeritus
Well, firstly, instead of calling it a "spot of red," why don't we talk about a particle? Now, I'm not sure what you're asking here. Are you saying that we don't know whether this particle exists or not and you want to find the probability that the particle is in this infinite space?

If so, then clearly if the particle exists, it exists in the infinite space with probability 1. If it doesn't exist, then this becomes a philosophical question. Either way, there's no way you can calculate, using maths, whether a particle exists or not.

I think this is far more of a philosophical question than it is mathematical. Unless, of course, I'm way off the mark in interpreting your question.

Edit: I've just re-read the clause re "substance" in your original post, and am now baffled as to firstly what you mean, and secondly the point of the discussion (especially in a mathematics forum!)

3. Jul 10, 2007

HallsofIvy

Staff Emeritus
If you have a finite number of possible outcomes, then you might declare them all to be "equally likely" and use the "uniform probability distribution" by default. That's often done.

However, if there are an infinite number of possible outcomes, then you MUST specify a specific probability distribution- there is no "uniform probability distribution" over infinite sets.

In other words, it is impossible to answer your question until you specify a probability distribution.

4. Jul 10, 2007

matt grime

There is no uniform _discrete_ probablity measure on an infinite set, Halls. The uniform continuous distribution does exist on a measurable (in the probablisitic sense - i.e. total measure =1) state space.

5. Jul 10, 2007

HallsofIvy

Staff Emeritus
Thanks for the correction- I mistakenly used "infinite set" when I meant a non-measurable set.

6. May 20, 2009

amitmandora

help in probability question

i have a problem, say a company X wants to increase sales, and plans to insert letters in it detergent packs, every pack contains one letter. now, there is infinite number of sets of alphabets, from A to Z, now, my question is... what is the average number of packets that a customer would buy to complete the word SPARK.