The claimed Monty Hall result becomes untrue with very subtle changes to the premises. For example, if the host blindly chooses randomly between remaining doors

The claimed Monty Hall result becomes untrue with very subtle changes to the premises. For example, if the host blindly chooses randomly between remaining doors, the resulting distribution is 1/3 contestant door choice correct (switch loses), 1/3 contestant and host both incorrect (switch wins), 1/3 contestant incorrect host correct (no switch can be offered).