. Do I need a probability distribution over crash frequencies? That was the solution the last time I was confused over Bayes, but I don't understand what it means to say "The probability of having the line, given crash frequency f", which it seems I need to know to calculate a new probability distribution.

What you want to find is P(line causes crash|observed crashes)

let's call

L: "line is cause of crash"
R: "crash is random" (so basically R = ~L)
O: "observed crash distribution"

p(L|O) = p(O|L)p(L)/p(O) = p(O|L)p(L)/(p(O|L) + p(O|R))

If your prior is that L and R are equally likely, this becomes

p(L|O) = 0.5/(1+p(O|R)/p(O|L))

... so you just need the ratio between p(O|R) and p(O|L).

To get that, you may want to compare two models, one where the crash occurs randomly with probability pr, one when the crash occurs only when the line is present with probability pl (pr and pl with some simple prior distribution like Jeffrey's prior, as DanielLC recommends).