Probability Problems and Simulation

There are a number of classic probability problems that challenge the intuition, both for students and for teachers. I have found that one way to overcome this intuition block is to write a quick simulation. A good example is the classic evil probability problem of the Monty Hall. The derivation of the solution is straightforward, but it is easy to convince yourself of the wrong answer. A quick simulation, like the one below, makes it clear: 1/3 of the time the host gets a choice with which door to open, and 2/3 of the time the host has no choice - with the other door having the prize. I find a numerical simulation helps to bolster my confidence in a mathematical analysis, especially when it is particularly unintuitive.

from random import randint
import random
turn=0
win=0
human=False

while turn<50:

   prize=randint(1,3)
   door_choices=[1,2,3]

   if human:
       your_first_answer=input('Which door %s? ' % str(door_choices))
   else:   # automatic
       your_first_answer=random.choice(door_choices)

   if prize==your_first_answer:  # happens 1/3 of the time
       door_choices.remove(your_first_answer)  # get the other two
       door_choices=sorted([your_first_answer,
                            random.choice(door_choices)])
   else:
       door_choices=sorted([prize,your_first_answer])

   if human:
       your_second_answer=input('Which door %s? ' % str(door_choices))
   else:   # automatic

       # always switch
        if door_choices[0]==your_first_answer:
           your_second_answer=door_choices[1]
       else:
           your_second_answer=door_choices[0]

   if your_second_answer==prize:
       print "You win!"
       win+=1
   else:
       print "You Lose!"

   turn+=1


print "Winning percentage: ",float(win)/turn*100