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Given for example, Hydrogen electron in ground state. What is probability to find that electron at certain distance (not interval of distances) from center of nucleus, for example at radial coordinate $r=0.5 \cdot 10^{-10}$ (and any angle)?

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are you asking for the probability for an exact radius and not the probability density for that radius? –  ACuriousJim Jun 24 '13 at 20:09
If so, the probability is identically zero –  ACuriousJim Jun 24 '13 at 20:12
In the interests of reducing the side chatter I'll concede the infinitesimal thing and delete my comments on the matter. Thanks to joshphysics, Jim and AlfredCentauri for correcting me. –  dmckee Jun 24 '13 at 23:26

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The probability of finding the particle exactly at a particular radial coordinate is zero whilst the probability of finding the particle in the infinitesimal neighborhood of that coordinate is infinitesimal.

$\int_r^r \rho(\tau)d\tau = 0$

$\int_r^{r + dr} \rho(\tau)d\tau = \rho(r)dr$

This follows from the fundamental relationship:

$f(x + dx) = f(x) + f'(x)dx$

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