I was trying to formalise the different methods of calculating probability in bertrand's paradox in logic, this is my attempt: \begin{align} \Phi \equiv x^2 + y^2 = 1\\ \Psi(m,c) \equiv y = mx + c\\ \chi(m,c) \equiv \exists x \exists y. \Phi \land\Psi(m,c) \\ \mu(m,c) \equiv \chi(m,c) \land \left(\frac{c}{\sqrt{m^2+1}}>0.5 \right) \end{align} I think what we need to evaluate is : $$ \Pr(\mu(m,c)|\chi(m,c))$$ Where $(m,c)$ are uniformly distributed (don't worry if you think evaluating this probability is meaningless (or not defined), I just want to formalise it ).
I realised that what I am modelling is none of the three methods.
Can anyone help me with formalising the three methods in this way ?