Introduction
The Monty Hall Problem is a puzzle named for Mr. Monty Hall, a game show host from the 60's and 70's. The problem can be summed up as follows: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?
In the video above there is detail on how to intuitively find the answer where you should always switch. Most people's intuitive answer is that after Monty Hall shows you the second door, you have a 50% chance of finding the sports car. While this is true (given two doors, one car), this way of looking at the problem ignores the fact that Monty Hall just revealed a goat to us. Remember that you had a 2/3 chance of getting a goat on your first pick. The following table I believe demonstrates the choices well:
Car in Door 1
|
Car in Door 2
|
Car in Door 3
|
Result in picking door #1 and not switching
|
Result if
switching
|
Car
|
Goat
|
Goat
|
Car
|
Goat
|
Goat
|
Car
|
Goat
|
Goat
|
Car
|
Goat
|
Goat
|
Car
|
Goat
|
Car
|
The program test
Hypothesis
Given enough iterations, the success rate of switching should tend to 66% given 3 doors.
Results
The following table was the output of the python script. as can be seen below, the percentage to 3 significant figures pans out to 66% victory rate for always switching, 33% for sticking to your guns, and an even 50-50 for randomly switching.
Runs: 10000Time: 0.72 seconds
Seed: 1
==========================
Always Switch:
Win: 6673
Loss: 3327
Percentage: 66 %
==========================
Never Switch:
Win: 3343
Loss: 6657
Percentage: 33 %
==========================
Sometimes Switch:
Win: 5041
Loss: 4959
Percentage: 50 %
==========================
