micro-sphere whose radius is equal to the Bohr radius $r_0$.

To estimate the above probability I will make an additional simplifying assumption. The wave function being not a constant function, the probability of presence will vary according to the location of the micro-sphere within the macro-sphere. However, since we are only interested in estimating a rough order of magnitude, we can assume that the wave function is a step-function, only taking two values: a constant non-zero value inside the macro-sphere, and a zero value outside of it.

With this simplification, we have everything we need to complete our estimation. For this, we have to remember that to calculate a probability of presence we have to integrate the squared module of the wave function over the region of interest. Considering the step-function hypothesis, this means that the probability that we seek will be proportional to the relative volume of the micro-sphere compared to the volume of the macro-sphere.

More exactly, given that the volume of a sphere is proportional to its radius to the cube, we obtain for the probability ${\cal P}$ that the hydrogen atom in question will be detected, after $1$ s, in a predetermined micro-sphere of radius $r_0$, 