| No. I am not a mathematician but I got it once I understood that this is not drawing coloured balls randomly from a bag (which is how all my school probability problems seemed to go). That is: - the setup of the system matters. - The state of the system at the point of the decision to switch matters. - The choices don’t get re-randomised. So the probabilities assigned to the original choice (and the remaining alternative) still count. If the host had closed a curtain over the stage and randomised the remaining doors, then it would be 50:50. But he didn’t. So you’re still in the probabilities of the original choice. One of the goats has been removed. The car and one goat remain: you know this for sure. You are being offered a door knowing that behind it must, necessarily, be the opposite of your original choice, and the probabilities have not been reset. If you originally picked the goat, that door absolutely has a car behind it. And there's a 2/3 chance you picked the goat originally. So by inference there's a 2/3 chance the door has a car behind it. You should switch. I didn’t get it until I had written a simulation to see it for myself though! |