1. Haifeng Luo

2. The Monty Hall Problem
Suppose you're on a game show, and you're given the
choice of three doors: Behind one door is a car; behind
the others, goats.
You pick a door, say No. 1, and the host, who knows
what's behind the doors, opens another door, say
No. 3, which has a goat.
He then says to you, "Do you want to pick door No. 2?”

3. Should you switch?

4. Background
First appeared in 1975, name comes from a TV show.
Original author claimed that you should switch.
Thousands of people disagreed, including many PhDs.

5. Two arguments
It is optimal to switch: probability of picking the
correct door initially is 1/3. So the other door has 2/3.
Or
It does not matter: the remaining two doors are equally
likely to contain the car.

6. Answer: SWITCH!
The key lies in how the host makes his decision.
If he intentionally opens a door with goat, then no
information is gain.
If the door is randomly chosen, then the two doors
have equal probability of containing the car.

7. Case 1
Player
picks 1
1/3
1/3
1/3
Car @ 1
Car @ 2
Car @ 3
Host shows 2 or 3
Host shows 3
Host shows 2
Host always shows goat!

8. Case 2
Player
picks 1
1/3
1/3
1/3
1/2
1/2
Shows 2 -- goat
Shows 3 -- goat
Shows 2 -- car
Shows 3 -- goat
Shows 2 -- goat
Shows 3 -- car
Car @ 1
Car @ 2
Car @ 3
✔
✔
✖
✖
Each case has 1/6
Probability

9. Back to Case 1
Player
picks 1
1/3
1/3
1/3
1/2
1/2
Shows 2 -- goat
Shows 3 -- goat
Shows 2 -- car
Shows 3 -- goat
Shows 2 -- goat
Shows 3 -- car
Car @ 1
Car @ 2
Car @ 3
✔
✔
✖
✖
1/6
1/6
0
1/3
1/3
0

10. Boys and Girls
Suppose a society really prefers boys over girls. Each
family tries their best to have a boy to continue the
male line.
Each couple will have one baby per year, and they stop
once they get a boy. A typical family may have, say, g, g,
g, g, b.
What’s the percentage of boys/girls after 100 years?

11. Still 50%!
Let’s say we have 1000 couples in the society.
1st year: 500 boys; 500 girls
2nd year: from the frustrated couples -- 250 boys; 250
girls
3rd year: as they keep going: 125 boys and 125 girls
And it continues …

12. The take-away
Our intuitions can be very misleading. Especially
regarding probability and statistics.
For more interesting examples, refer to the book
Thinking, fast and slow by Daniel Kahneman.