Malus' law ($I=I_0{\cos }^2\theta$)

An ideal linear polarizer transmits the fraction ${\cos }^2\theta$ of incident linearly polarized light when its axis makes angle $\theta$ to the polarization. For a single photon, the corresponding conditional transmission probability is $|⟨\theta |0⟩{|}^2={\cos }^2\theta$, by the Born rule; rejection has probability ${\sin }^2\theta$. The classical field calculation projects its electric-field amplitude onto the transmission direction and squares that amplitude to obtain intensity. This is a conditional probability given the incident polarization and is not automatically the joint probability of passing one analyzer and failing another.