Random double

[Ivan Kazmenko]

On Wednesday, 24 April 2013 at 10:46:57 UTC, Andrea Fontana wrote:
> What's the probability to guess a precise number in [0..1]? I 
> think is 0 as long as you have infinite numbers.

Right.

> What's the probability to guess a interval in [0..1]? I think 
> it's the interval size.

Right again.

> I mean that a random real number is not valid mathematically 
> too. In any given real interval there are infinite numbers, how 
> you can choose a number in an infinite (and non-numerable!) 
> interval? I think you always need some sampling.
>
> Am I wrong?

In a sense, yes.  A continuous probability distribution is 
well-defined.  In short, as you pointed out, you can coherently 
define the probabilities to hit each possible segment and get a 
useful mathematical object, though the probability to hit each 
single point is zero.  It's no less strict than a typical high 
school definition of an integral.  See the link for more info: 
http://en.wikipedia.org/wiki/Probability_distribution#Continuous_probability_distribution 
.