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Misconceptions in Probability
Haifeng Luo
The Monty Hall ProblemSuppose 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?”
Should you switch?
BackgroundFirst appeared in 1975, name comes from a
TV show.Original author claimed that you should
switch.Thousands of people disagreed, including
many PhDs.
Two argumentsIt 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.
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.
Case 1
Playerpicks 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!
Case 2
Playerpicks 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
Back to Case 1
Playerpicks 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
Boys and GirlsSuppose 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?
Still 50%!Let’s say we have 1000 couples in the society.
1st year: 500 boys; 500 girls2nd year: from the frustrated couples -- 250
boys; 250 girls3rd year: as they keep going: 125 boys and
125 girls
And it continues …
The take-awayOur 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.
