Result when you pick door #, then switch choice

Result when you pick door #, and don’t switch

↓ probability that the Auto is behind any particular closed door you didn’t pick
 (N-1)/N(N-M-1) =1/((N-M) )+M/N(N-M)(N-M-1)
    your door ↑            ↑ probability it is behind any closed door as assumed by a new person

↓ probability that the Auto is behind any particular closed door you didn’t pick
          =       (Monty’s opened one door)
                                  ↑ probability it is behind any closed door including
                                     yours as assumed by the new person

↓ probability that the Auto is behind any particular closed door other than the one you’ve now picked