Sometimes the universe is just weird. Not weird in the things-are-really-completely-random way, but in the profound-truth-that-I-can’t-possibly-understand way. You know, Banack-Tarski Paradox weird. Euler’s Identity weird. Weird like Monty Hall weird.
Ok, I don’t really mean Monty Hall, I mean the Monty Hall Problem named after him. But I’m sure he is weird. Everything that was popular in the 70s was weird.
Anyway, the Monty Hall Problem. It’s pretty weird. The basic premise goes like this: you get to choose 1 of 3 doors. 1 of the 3 doors has a new car behind it, the other 2 have goats. After you’ve made your choice, Monty Hall shows you a goat behind one of the doors, and asks if you’d like to switch doors to the other remaining unopened door. Is there a statistical difference between your two choices? (For a precise definition, see the Wikipedia article.
I won’t keep you in suspense. You’ve got a 2 in 3 shot at it if you switch, and only a 1 in 3 shot if you don’t. There really is a difference, as the myriad proofs on Wikipedia show. But it’s just a weird answer. Our gut tells us you’ve got a 50/50 shot either way.
A while back, I argued with a friend about this for awhile. She was absolutely convinced that it was 50/50. We went back and forth for awhile and didn’t really get anywhere (my verbal exposition was not up to snuff). Today I read a New York Times article that talks about the problem and it reminded me of her insistence. Sometimes there’s just nothing like empirical evidence. So I whipped up a little script to run through a large sample set for this problem.
Here’s the results from 2 runs:
DO NOT SWITCH DOORS: 1,000,000 trials conducted
Results:
Pick Distribution:
Door 1: 332588
Door 2: 333606
Door 3: 333806
Goat Distribution:
Door 1: 333507
Door 2: 333938
Door 3: 332555
Correct: 333134, 33%
Incorrect: 666866, 67%
=====
SWITCH DOORS: 1,000,000 trials conducted
Results:
Pick Distribution:
Door 1: 332871
Door 2: 333808
Door 3: 333321
Goat Distribution:
Door 1: 333779
Door 2: 332836
Door 3: 333385
Switch Distribution:
Door 1: 333350
Door 2: 333356
Door 3: 333294
Correct: 666571, 67%
Incorrect: 333429, 33%
========================================
DO NOT SWITCH DOORS: 1,000,000 trials conducted
Results:
Pick Distribution:
Door 1: 334011
Door 2: 333256
Door 3: 332733
Goat Distribution:
Door 1: 332870
Door 2: 333830
Door 3: 333300
Correct: 333308, 33%
Incorrect: 666692, 67%
=====
SWITCH DOORS: 1,000,000 trials conducted
Results:
Pick Distribution:
Door 1: 333526
Door 2: 333202
Door 3: 333272
Goat Distribution:
Door 1: 332794
Door 2: 333637
Door 3: 333569
Switch Distribution:
Door 1: 333680
Door 2: 333161
Door 3: 333159
Correct: 666985, 67%
Incorrect: 333015, 33%It works out. But it’s still non-intuitive. I just love that the world works like that sometimes. If things were always real simple, the entire Universe would be completely knowable, and that would be boring. So rejoice when things are weird.