# From a cross section to a probability

This is homework problem: Given the cross-section of a neutrino-electron scattering, what is the probability for a solar neutrino to scatter with a electron as it goes through the center of the Earth?

I'm quiete confused: the dimensions of a cross section are of $L^{-2}$, to get a probability one has to multiply with the area of a section of the earth, if one had to deal with a flux of particles covering the complete area. But here there is only one particle!

I thought a cross-section would be $cm^2$, not $cm^{-2}$.
The information you really lack is probably the macroscopic cross section. That is, $\Sigma = n \sigma$, where n is the number density of the target objects, in number per cubic centimeter. Thus, this gives a value of inverse length units.
$$P(0) = 1.0 \\ \frac{dP}{dx} = - \Sigma P(x) \\ P_s(x) = 1 - P(x)$$
For justification, $\Sigma \Delta x$ really is a sort of probability, so just put that in calculus terms in a way that makes sense to you.