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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!

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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.

Using this, you can readily speak in terms of the cumulative probability distribution (CDF). I would start with the probability of being uncollided. Once you have that, unity minus that is the probability that it is collided. For instance, you might use this as your start point:

$$ P(0) = 1.0 \\ \frac{dP}{dx} = - \Sigma P(x) \\ P_s(x) = 1 - P(x) $$

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.

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