 If the player picks door 1 and the host's preference for door 3 is "q", then the probability the host opens door 3 and the car is behind door 2 is while the probability the host opens door 3 and the car is behind door 1 is .These are the only cases where the host opens door 3, so the conditional probability of winning by switching "given the host opens door 3" is which simplifies to .Since "q" can vary between 0 and 1 this conditional probability can vary between and 1.