Let's see what happens when let's make a deal is played with four doors. a prize is hidden behind one of the four doors. then the contestant picks a door. next, the host opens an unpicked door that has no prize behind it. the contestant is allowed to stick with their original door or to switch to one of the two unopened, unpicked doors. the contestant wins if their final choice is the door hiding the prize.
lets make the same assumptions as in the original problem:
the prize is equally likely to be behind each door.
the contestant is equally likely to pick each door initially, regardless of the prizes location.
the host is equally likely to reveal each door that does not conceal the prize and was not selected by the player.
use the four step method to nd the following probabilities. the tree diagram may become awkwardly large, in which case just draw enough of it to make its structure clear.
(a) contestant stu, a sanitation engineer from trenton, new jersey, stays with his original door. what is the probability that stu wins the prize?
(b) contestant zelda switches to one of the remaining two doors with equal probability. what is the probability that zelda wins the prize?
(c) now let's revise our assumptions about how contestants choose doors. say the doors are labeled a, b, c, and d. suppose that carol always opens the earliest door possible (the door whose label is earliest in the alphabet) with the restriction that she can neither reveal the prize nor open the door that the player picked.
this gives contestant mergatroid just a little more information about the location of the prize. suppose that mergatroid always switches to the earliest door, excluding his initial pick and the one carol opened.
what is the probability that mergatroid wins the prize?