I am wondering how to calculate how far a neutrino would have to pass through a substance for it to have a probability P of interacting at least once. Water, for instance, has a density of 1 g / cm^3; using Avogadro's number I think this means that there is about 6.02 x 10^29 protons and/or neutrons per 1 m^3; as a proton or a neutron is made up of three quarks, this means that for water there is about 1.806 x 10^30 quarks per m^3. I know that a neutrino has to get very close (~10^-18 m) to a quark to interact with it, and that very few of the neutrinos that make it into the radius of some arbitrary quark actually do interact with it (~1 in 10^12). How could I use this information to (VERY) roughly estimate how far some random neutrino would have to pass through water before it would interact with a quark? Or, say, how long it would have to pass before it had a 60% chance of interacting? Any help, hints or guidance to solving this problem would be appreciated. I have read (here, for example: http://www.physics.usyd.edu.au/hienergy/forces_and_neutrinos.html) that a neutrino would have to travel ~9.5 x 10^17 m through pure water before interacting, but I haven't been able to find any explanation of how this may be reasoned out.