The Monty Hall Problem

the Pledge

Imagine you’re in a game-show. You have managed to blaze through the quarter-finals and semi-finals and end up as one of the two finalist contestants at the threshold of receiving the Grand Prize of 1 Million Dollars!

MHP all closed

Your final challenge is a simple guessing game. There are three doors: one of which contains the Grand Prize, and the remaining two contain a beautifully groomed and domesticated goat each. You must guess which door contains the Grand Prize. If you guess correctly, you receive the Grand Prize and are declared official winner of the game show. If you guess incorrectly, you will receive the goats, leaving the Grand Prize to your opponent.

MHP 3 open GMG

You have no way of knowing which door conceals which item. Whichever door you pick, you will receive the prize behind it.

The game-show host asks you to pick a door. Let us say you had a good feeling about Door A and picked it. Here’s the twist. Before the game-show host reveals the contents behind Door A, he reveals to you the content of one of the remaining two doors. The game-show host will always reveal the location of one of the goats. Let’s say he opened Door B, which contained a goat.

MHP AGC

Then game-show host offers you a second chance: do you want to stick with your initial choice, or would like to switch doors?

MHP A'Host'C

The question is:

Should you swap?

Should you stick with your original choice?

Or does it really make a difference?

the Turn

If your answer is  that it doesn’t really make a difference whether you switch doors or not, then you are unfortunately, very very wrong. Surprising isn’t it? It in fact does make a difference, a very large one in fact, as you actually double your chances by switching from your initial choice. Don’t believe me?

Never fear, Mathematics is here! 

the Prestige

Assuming you are among the many who believed it didn’t make a difference (if you’re a smarty-pants and chose correctly to swap doors, then go be smart some place else…), then you probably made the false assumption that there was a 50% chance that the Grand Prize would be in Door A as well as Door C.

MHP SG-G-SM MHP SM-G-SG

Here’s an example.

We will use the same arrangement, where Door A contains a Goat, Door B contains a Goat, and Door C contains the Grand Prize.

MHP G-G-M

Let us set some cases:

1. You pick Door A [it contains a Goat], the Host shows you Door B [it contains a Goat]

MHP PSG-G-SM

2. You pick Door B [it contains a Goat], the Host shows you Door A [it contains a Goat]

MHP G-PSG-SM

3. You pick Door C [it contains the Grand Prize], the Host shows you Door A/B [it contains a Goat]

MHP SG-G-PSM

Let’s count the number of times you win if you do not switch doors and remain with your initial choice.

1. You pick Door A [it contains a Goat], the Host shows you Door B [it contains a Goat], you remain with Door A.

MHP PG-G-SM

YOU LOSE

2. You pick Door B [it contains a Goat], the Host shows you Door A [it contains a Goat], you remain with Door B.

MHP G-PG-SM

YOU LOSE

3. You pick Door C [it contains the Grand Prize], the Host shows you Door A/B [it contains a Goat], you remain with Door C.

MHP SG-G-PM

YOU WIN

Therefore:

  • P(Lose if you do not switch) = 2/3
  • P(Win if you do not switch) = 1/3

Let’s count the number of times you win if you switch doors rather than remaining with your initial choice.

1. You pick Door A [it contains a Goat], the Host shows you Door B [it contains a Goat], you switch to  Door C.

MHP PSG-G-PM

YOU WIN

2. You pick Door B [it contains a Goat], the Host shows you Door A [it contains a Goat], you switch to  Door C.

MHP G-PSG-PM

YOU WIN

3. You pick Door C [it contains the Grand Prize], the Host shows you Door B [it contains a Goat], you switch to  Door A.

MHP PG-G-PSM

YOU LOSE

Therefore:

  • P(Lose if you switch) = 1/3
  • P(Win if you switch) = 2/3

You can also come up with the same answer by drawing a tree diagram,

MHP Tree Diagram

or using Conditional Probability calculations,

MHP CP

As you can see, the Probability of Winning doubles when you decide to switch doors and the Probability of Losing halves.

This problem is known as the Monty Hall Problem or Monty Hall Paradox, and is named after the host of the TV Game Show “Let’s Make a Deal”. This problem is recognized for its counter-intuitive result – a Veridical Paradox.

For many years, it has baffled people, especially renowned mathematicians, for its paradoxical solution. Numerous simulations have been created to test whether you are more likely to win if you switch doors. The New York Times created a Flash Simulation on their website that allows you to test for yourself the Monty Hall Problem.

The Mythbusters and many other science shows have all tried their hand at this problem, and switching has always proven to be the most profitable choice.

The most important lesson to take from the Monty Hall Problem is to realize that our gut/instinctive feeling is not always correct, and that we must always be skeptical of everything.

 

Works Cited

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