Hydrogen ground-state nuclear-volume probability ($P(r<r_p)\simeq 4r_p^3/(3a_0^3)$)

For the point-Coulomb ground state, radial probability density is $4r^2{e}^{-2r/a_0}/a_0^3$. Integrating to $r_p$ gives $1-e^{-2u}(1+2u+2u^2)$, with $u=r_p/a_0$, whose small-$u$ expansion is the displayed result. A finite proton charge distribution changes the interior potential, and a proton radius is not literally a sharp material boundary; the expression is an estimate of a small overlap region.