Question

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.

Let’s make the same assumptions as in the original problem:

(a) The prize is equally likely to be behind each door.
(b) The contestant is equally likely to pick each door initially, regardless
of the prize’s location.
(c) The host is equally likely to reveal each door that does not conceal
the prize and was not selected by the player.

Find the following probabilities. If the tree diagram is too large, you can
draw just enough of it for the structure to be clear.

(a) Contestant Stu 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?

Now let’s revise our assumptions about how contestants choose doors. Say
the doors are labeled A, B, C, and D. Suppose that the host always opens
the earliest door possible (the door whose label is earliest in the alphabet)
with the restriction that the host can neither reveal the prize nor open the
door that the player picked. This gives contestant Priscilla just a little
more information about the location of the prize. Suppose that Priscilla
always switches to the earliest door, excluding her initial pick and the one
the host opened.

(c) What is the probability that Priscilla wins the prize?

Answer 1

Solution:

a)

b)

c)

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