probability in an infinite set Let's say we have a truly random real number between 0 and 1 inclusive both end points.
given that we have an infinite number of number between 0 and 1 what is the probability of a number picked being exactly 1/2 ? Is it zero? Is the probability of a number less then .4 as I would expect 40%? What's the probability of of a rational number? irrational number? transcendental number? My intuition tells me you would pick a transcendental number every time but that's not based on logic it's a guess. 
Finally is the question itself valid? Can we even have a set of all the numbers between 0 and 1 and then pick one at random? Does any branch of math allow this? clearly we can do this with any finite set but math comes up with some odd answers regarding infinite sets.  example. 
I suspect that the answer to this question will no be meaningful in a practical sense but that's not what pure math is about anyway. it's about laying the ground rules building up from there and seeing what happens. Right?
 A: Yes, the probability of picking any one number from a uniform distribution on $[0,1]$ is zero. You can make this rigorous, but hopefully this can fit your intuition. 
The probability of picking a number less than $0.4$ is also 40% as you stated.
You can define the so-called pdf $f(x) = 1$ and then we can define the probability of picking a number in any interval $[a,b] \subset [0,1]$ by
$$Pr([a,b]) = \int_a ^b 1\rm{d}x $$
Now, when you're dealing with sets that aren't intervals, it becomes a little bit more complicated.  You can show that there are many many more irrational numbers than rational numbers (and similarly many more transcendental numbers).  It turns out the probability of picking a transcendental number is 1 (similarly an irrational number). We can actually make this rigorous with the above formalism:
$$\int_{\mathbb{[0,1] -Q}} 1 \rm{d}x = 1$$
where we have used the Lebesgue integral.
As for practicality, such formalism is useful in electrical engineering and signal processing, as well as many other fields.
A: With the uniform distribution, the probability to pick a number in any measurable subset of $[0,1]$ equals the measuer of that set. Hence the probability for any particular number is $0$; the probability for a number $\in[0,0.4]$ is $0.4$; the probability of a rational is $0$; irrational $1$; transcendental $1$. So even if you happen to actually pick the number $0.89235123951661\ldots$ - the probability to pick it was $0$.
In a way, the question is indeed invalid, as we never really pick a number. In a way, you could say that probability theory is always only about "what if we picked a number?" (Formally, it is only about measureable sets and functions)
