Probability question (easy to ask, hard to answer)

[ultrafilter]

Oops. Leaned on the tab key and hit the wrong button. Let’s try that again.

*Originally posted by jawdirk *
Jabba, that makes sense, but it leaves me with the question of whether probability is the right word to use when speaking of continuous random variables. Suppose I ask the question: what is more probable, rolling a 7 on a six-sided die or generating a rational value from a random selection of a real number on the range [0,1]? Are you going to say that they are equally probable or are you going to say that the question is meaningless because I am talking about two different sorts of probability?

Both events have zero probability, but one can happen, and the other can’t.

**It would seem far less confusing to just say that the probability of generating a rational number over real [0,1] is undefined. I’m not questioning that the Lebesque measure of the rationals is 0, I’m questioning that the Lebesque measure corresponds to probability, which as this example demonstrates, really only fits our intuitions in the context of discrete random variables or ranges within continuous random variables. **

Why should it match our intuition in every setting? We shouldn’t declare things undefinable just cause they’re weird, cause we’d lose a lot of interesting stuff.

And to answer your question, probability is nothing but a measure on a set, constrained by the axioms of probability theory (link coming soon). Measure theory is weird, so there’s no reason to expect that probability theory won’t be weird too.

[ultrafilter]

*Originally posted by Tretiak *
**So, have we answered the OP yet? **

Yes. It’s undefined, and it’s counterintuitive.

[Jabba]

originally posted by Ultrafilter
Measure theory is weird

I think this is the key to the intuitive difficulties: our intuitions ( or mine, at least) are not reliable in the case of measurable sets. Cosider the set of rational numbers in [0,1], the source of the difficulty in this case. Between any two distinct reals there is a rational number. There are infinitely many rational numbers between 0.001 and 0.002. On the other hand it is possible to cover the set of rationals with a sequence of intervals whose total length is less than 0.0000000001. My intuition has enormous difficulty in reconciling these two facts.