1. Let’s Make a Deal
February 21, 2012

2. We Begin With
player faces a choice of 3 doors–behind one is the prize
after initial guess, Monty Hall reveals a Zonk! behind one of
two remaining doors
contestant can switch curtains after the reveal
question: does it pay to switch?

3. Further ...
Let G be the door the prize is behind ∈ (1, 2, 3)
Let C be the first choice of the contestant ∈ (1, 2, 3)
Let M be the door that Monty reveals ∈ (1, 2, 3)
Without loss of generality, let the ratio of winning by
switching to not switching be represented by this situation:
P(G = 3|M = 2, C = 1)
, (1)
P(G = 1|M = 2, C = 1)

4. Rewriting ...
P(W |S) P(G = 3|M = 2, C = 1)
=
P(W |N) P(G = 1|M = 2, C = 1)
P(M=2,C =1|G =3)∗P(G =3)
P(M=2,C =1)
= P(M=2,C =1|G =1)∗P(G =1)
P(M=2,C =1)
P(M = 2, C = 1|G = 3) ∗ P(G = 3)
=
P(M = 2, C = 1|G = 1) ∗ P(G = 1)
P(M = 2|C = 1, G = 3) ∗ P(C = 1|G = 3)
= (2)
P(M = 2|C = 1, G = 1) ∗ P(C = 1|G = 1)

5. Simplifying
P(W |S) P(M = 2|C = 1, G = 3)
= (3)
P(W |N) P(M = 2|C = 1, G = 1)
numerator = 1: Monty must choose door 2 if contestant chooses 1
and prize is behind 3
denominator depends on Monty’s behavior; if he chooses randomly,
then denominator= 1/2, ratio = 2–as in the simpler calculation,
switching is about twice as likely to win as not
we don’t know how Monty chooses, but if there is any randomness
to Monty’s choice, denominator <1–better off switching