rapid habits

Monty Hall paradox: see for yourself

The Monty Hall paradox/problem goes something like this:

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?” Is it to your advantage to switch your choice?

The thing that’s always bugged me about this common phrasing (and the accompanying solution) is the assumption that the host always opens a door, revealing a goat. This isn’t exactly evident from the problem definition; we only know that the host knowingly revealed a goat. The host could have been doing so maliciously; revealing a goat only when the competitor chooses a car the first time around in hopes to lure her away from the prize. And I’m not just reading too much into the puzzle; we need to phrase this problem as a repeatable scenario in order to have any notion of probabilistic strategy. The hosts behavioral pattern (random, predetermined or whatever) is a key component to this repeatable scenario.

Still, the switching strategy does yield a higher probability of scoring a sweet ride, when we know that the host will always reveal a goat. Just to put a stake through the heart of my doubt about Monty Pyth Hall’s true benevolence, I wrote something in Python to settle this once and for all. The results are not deterministic because I don’t seed the random number generator with a constant before execution, but the manual runs all yielded values within reasonable ranges of one another. The empirical results are as the common probabilistic solution suggests:

Here’s the source, try it out if you don’t believe me:

#!/usr/bin/python
# Mustafa Paksoy, April 08
# Usage: ./monty.py [iterations] [debug (optional)]
# Empirically evaluate the outcome of the Monty Hall problem.
from random import choice
from sys import argv

def safeRem(list,num):
  if num in list:
    list.remove(num)

correct_stay=0
correct_switch=0
tries=int(argv[1])
debug=len(argv)>2
for i in range(tries):
  car=choice(range(3))
  pick=choice(range(3))
  others=range(3)
  safeRem(others, car)
  safeRem(others, pick)
  monty=choice(others)
  others=range(3)
  others.remove(monty)
  others.remove(pick)
  newpick=others[0]
  if debug: print 'car', car, 'pick', pick, 'monty', monty, 'newpick', newpick,
  if newpick == car:
    correct_switch+=1
    if debug: print 'switch'
  elif pick == car:
    correct_stay+=1
    if debug: print 'stay'
  else:
    raise ValueError("you've got a bug")

print 'tries:', tries
print 'stay correct:', correct_stay, float(correct_stay)/tries
print 'switch correct:', correct_switch, float(correct_switch)/tries