Andrew Fraser DBA

Of Goats and Porsches: A Monty Hall Paradox Simulator

The Monty Hall Paradox can’t be right, or so I thought.

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?

So I wrote the below simulator to find out. It uses the Monte Carlo Technique, which is a nice way of solving math problems numerically.
The results from my run are pretty conclusive:

Twitchy winners : 653
Twitchy losers : 347
Stubborn winners : 318
Stubborn losers : 682

So it really does pay to switch doors when the host gives you the choice. Which is not what I had expected – and means it’s time for me to go eat some humble pie 🙁 .

Code listing below:

#!/bin/ksh
#######################################################
# monty_hall.sh
# Andrew Fraser ( http://andrewfraserdba.com )
# January 2007
# Simulator for the Monty Hall 'paradox'. See http://en.wikipedia.org/wiki/Monty_Hall_problem for background info
# Usage:
#  monty_hall.sh <no_iterations> <twitchy|stubborn>
#  e.g.: monty_hall.sh 1000 twitchy
# Parameters:
#   1 - NO_ITERATIONS - number of times to run simulation (a few hundred is reasonably fast on most machines)
#   2 - CONTESTANT_TYPE - twitchty = moves doors when given chance to move : stubborn = stays put when given chance to move
# Notes:
#   ksh RANDOM built in function is not the best random function in existence, although there are patches that resolve that.
#   RANDOM returns a number between 0 and 32768. The three doors here are named "0", "1", and "2", achieved by dividing by 10924
#
#   A good idea is to redirect the output for larger runs, and grep out the results. E.g.:
#     monty_hall.sh 1000 twitchy > twitchy.out
#     monty_hall.sh 1000 stubborn > stubborn.out
#     echo Twitchy winners : `grep Congratulations twitchy.out|wc -l`
#     echo Twitchy losers : `grep dear twitchy.out|wc -l`
#     echo Stubborn winners : `grep Congratulations stubborn.out|wc -l`
#     echo Stubborn losers : `grep dear stubborn.out|wc -l`
#######################################################

i=0

NO_ITERATIONS=$1
CONTESTANT_TYPE=$2

while [ $i -lt $NO_ITERATIONS ]
do

CONTESTANT_DOOR1=`echo $(( RANDOM / 10924 ))`
echo CONTESTANT_DOOR1 is $CONTESTANT_DOOR1
PORSCHE_DOOR=`echo $(( RANDOM / 10924 ))`
echo PORSCHE_DOOR is $PORSCHE_DOOR

if [ $CONTESTANT_DOOR1 -eq $PORSCHE_DOOR ]
then
RIGHT_FIRST_TIME=yes
#host now uncovers one (at random) of the two remaining doors to show a goat...
ZERO_OR_ONE=`echo $(( RANDOM / 16384 ))`
if [ $PORSCHE_DOOR = 0 ]
then
let UNCOVERED_GOAT_DOOR=1+ZERO_OR_ONE
elif [ $PORSCHE_DOOR = 1 ]
then
if [ $ZERO_OR_ONE = 0 ]
then
UNCOVERED_GOAT_DOOR=0
else
UNCOVERED_GOAT_DOOR=2
fi
elif [ $PORSCHE_DOOR = 2 ]
then
UNCOVERED_GOAT_DOOR=$ZERO_OR_ONE
fi
echo UNCOVERED_GOAT_DOOR is $UNCOVERED_GOAT_DOOR
else
RIGHT_FIRST_TIME=no
#host now uncovers the only other goat door to show a goat...
let UNCOVERED_GOAT_DOOR=3-CONTESTANT_DOOR1-PORSCHE_DOOR
echo UNCOVERED_GOAT_DOOR is $UNCOVERED_GOAT_DOOR
fi

if [ $CONTESTANT_TYPE = twitchy ]
then
# contestant moves to the other uncovered door, maybe cos she read something about that being good strategy on wikipedia...
if [ $RIGHT_FIRST_TIME = yes ]
then
echo ":( Oh dear, you won a goat, what a shame, you were right first time, if only you had stayed where you were! :("
else
echo ":) Congratulations! You've won a porsche :)"
fi
else
# contestant stubbornly stays where she is...
if [ $RIGHT_FIRST_TIME = yes ]
then
echo ":) Congratulations! You've won a porsche :)"
else
echo ":( Oh dear, you won a goat, what a shame, you should have moved door. :("
fi
fi

echo

let i=i+1

done

For the actual run I did, with results pasted above, I ran:

monty_hall.sh 1000 twitchy > twitchy.out
monty_hall.sh 1000 stubborn > stubborn.out
echo Twitchy winners : `grep Congratulations twitchy.out|wc -l`
echo Twitchy losers : `grep dear twitchy.out|wc -l`
echo Stubborn winners : `grep Congratulations stubborn.out|wc -l`
echo Stubborn losers : `grep dear stubborn.out|wc -l`