Quote by Bob_for_short
If an average is equal to zero, it does not mean that the zero is most probable point.
Any point in 3D is characterized with two angles and the distance from the origin. So any probability dw to find a particle in an elementary volume dV is the product of the wave function squared ψ(r)^{2} and dV=r^{2}dr⋅d(cos(θ))⋅dφ. Due to the factor r^{2} this probability tends to zero at r=0. So the most probable distance is not zero but ~a_{0}.

I never said the most probable
distance was zero. The most probable distance is indeed the Bohr radius, and the most probable point is r=0. These quantities have different dimensions; the radial wave function and its square has the dimension of a 3d point  an infinitesimal volume element. The radial probability distribution, which you're referring to, is onedimensional quantity  an infinitesimal line segment.
What's so difficult to grasp about this? If I paint a set of spheres with an amount of paint that's exp(r) per area unit (and somehow infinitely thin regardless of amount), then the
point that has the greatest amount of paint isn't the same thing as the sphere that has the maximum amount of paint on it.