The ground state wave function for the hydrogen atom is y(r, q, f) = (pa_o³)^-1/2 exp(-r/a_o).

a) Find the probability that the electron occupies a small volume DV = (a_o)³ at three radii:   r = 0, a_o, and 2a_o.
A plot of the probablity density (in units of probability per a_o³) is shown below.

The probability of occupying DV near r = 0 is    
The probability of occupying DV near r = a_o is    
The probability of occupying DV near r = 2a_o is    

b) Find the probability that the electron occupies a thin spherical shell of thickness Dr = a_o, at three radii:  r = 0, a_o, and 2a_o.
A plot of the radial probablity density (in units of probablity per a_o) is shown below.