Question

1) The Monty Hall problem is a counter-intuitive statistics puzzle:
- There are 3 doors, behind which are two goats and a car.
- You pick a door (call it door 1). You’re hoping for the car of course.
- Monty Hall, the game show host, examines the other doors (2 & 3) and opens one
with a goat. (If both doors have goats, he picks randomly.)
Here’s the game: Do you stick with door A (original guess) or switch to the
unopened door? Does it matter?
a) Describe the sample space in case you stick with your original choice (either door
1, 2, or 3). Use C for finding a car, and G for finding a goat.
b) Describe the sample space in case you switch to the unopened door.
c) What are the odds of finding a car upon opening the door of your choice in the
case you stick with your original choice?
d) What are the odds of finding a car upon opening the door of your choice after you
have switched to the unopened door?

Answer 1

Door with car	Door picked	Stay	Switch
A	A	C	G
A	B	G	C
A	C	G	C
B	A	G	C
B	B	C	G
B	C	G	C
C	A	G	C
C	B	G	C
C	C	C	G

(a) The sample space for sticking to the original choice is the third column in the table.

C = 3/9 = 1/3, G = 6/9 = 2/3

(b) The sample space for switching the choice is the fourth column in the table.

C = 6/9 = 2/3, G = 3/9 = 1/3

(c) Odds ratio = P/1-P , where P is the probability of the event for which odds are required.

So for odds for finding car if stuck to original choice is (1/3)/(1-1/3) = 1/2

(d) Odds for finding a car if switched the door is (2/3)/(1-2/3) = 2