Probability of neutrino-electron scattering

[Rick88]

Homework Statement

Ok, I'm given a formula for the cross section of the scattering and I've been told that the detector is a column of water of depth 10m. I need to find the probability of the scattering within the detector.

Homework Equations

σ = E(in MeV) x 1.5x10^-44 cm²

The Attempt at a Solution

I know the cross section is actually a probability in particle physics, but I just don't see how to use the depth of the detector.
A ratio, maybe?

 

[vela]

The cross section is a measure of the likelihood (not a probability strictly speaking) that one electron and one neutrino will interact. When the neutrino goes through the column of water, it will encounter many electrons, and each time, there's a chance it will scatter. You need to sum the probabilities of all those encounters to find the total probability of scattering within the detector.

 

[Rick88]

Yeah, that's what I thought, but I am confused about how to find the total probability.
An integral or what?

 

[vela]

It's been a long time since I took particle physics, so I may be totally off here. I think you can just take the cross section and multiply it by the length of the detector, and if an electron is inside this volume, it will interact with the neutrino. So it comes down to figuring out what the probability is of finding an electron inside this volume, which you should be able to determine from the density of water.