Hello how to show the following
You are given the choice of 3 doors. Behind one is a car and the other two are goats. You pick a door uniformly at random say 1, and Monty opens another door, say 3 which has a goat. Monty asks "do you want to switch your choice to door 2?" If we have the following modified assumptions:
*The car was put behind a uniformly chosen random door.
*Monty knows where the car is and always opens a door with a goat behind it.
*When Monty has a choice(i.e. the contestant picked the door with the car behind it), he chooses the door with the larger number with probability 3/4.
Show that if the contestant switches, she wins with probability 4/7 not 2/3.
Thanks a lot!