Wednesday, May 5, 2010

Monty Hall Problem: an intuitive explanation

The Monty Hall Problem is a probability puzzle based on a TV show. Here's the puzzle:
Suppose that you are on a game show, and the host shows you three doors. He tells you that behind two of the doors are goats, and behind one of the doors is a brand new luxury car. He asks you to pick one of the three doors, and if you picked the door with the car behind it, you keep the car. You pick a door (say door #1). The host, knowing which door has the car behind it, walks over to a different door (say door #2) and opens it to reveal a goat. He then offers you to a chance to change your mind (and switch to door #3). Should you make the switch?
If you haven't heard the puzzle a billion times already, think for a bit before reading on.

Here's the answer: you should switch. I'll give two explanations as to why. The first one will (hopefully) appeal to your intuition, and the second one will be an argument using probability.

The Intuitive Explanation

Let's change the game for a bit. Suppose instead of only 3 doors, we have 100 doors: with 99 goats still only one car behind the doors. After you pick a door (say door A), the host opens 98 doors to reveal 98 goats, only leaving one other door (say door B) closed. In this case, would you choose to switch (to door B)? Again think about this first before reading on.

The Probabilistic Explanation

Here's how you might have reasoned about the previous scenario: the only case where switching to door B is not beneficial is when you choose the right door the first time. That only has a 1% chance of happening.

The same reasoning applies to the 3 doors scenario. The only case where switching would not help you is when you choose the door with the car behind it the first time. There's a 33% chance of that happening, and a 66% chance of picking the wrong door. Thus you will double your probability of winning the car if you decide to switch.

End of Entry