Firstly, it depends what kind of random number it is. If you flip two coins(0/1) and take the average (technically in [0,1]), you'll get a very different kind of random number than if you say every number [0,1) is equally likely to be chosen.
Secondly, looking at the three you mention, if you're interested in [0,1) exponential or gaussian would be bad models as they take values in $[0, \infty)$ and $(-\infty, \infty)$ respectively. Uniform(0,1) would be the natural choice, but as above, other brands are available.
The mass of the function in [0.5, 0.6] will entirely depend what distribution is used. If Uniform(0,1) then 0.1.