Monty Hall Revisited

Would You Switch or Stick?

In a Nutshell

It’s a game show, and you have a chance of winning a brand gleaming new red sports car. All you need to do is choose which of three identical doors leads to the car. They are labelled 1, 2 and 3.

You choose Door 1.

The host, name of Monty Hall, knows where the car is, and opens Door 3, which he knows leads to a goat. He might equally well have opened Door 2, if that concealed the goat. So Door 3 is now open.

No problem – you’d prefer the car!

There are two unopened doors left. Door 1, which you chose, and Door 2. It looks like you now have a 50/50 chance of winning the car.

At this point, Monty offers you a choice, to stick with Door 1 or switch to Door 2. He always makes this offer, so there’s no clue in the offer.

But does it matter what you do – switch or stick?

It certainly looks as though it shouldn’t. There are two doors left and the car must be behind one of them. Surely it’s now 50-50?

It isn’t!

Marilyn vos Savant

The problem became famous when it appeared in Marilyn vos Savant’s column in Parade magazine. She said you should switch.

Switching, she said, gives you a 2 in 3 chance of winning the car. Staying with your original choice gives you only a 1 in 3 chance.

How come?

When you first choose Door 1, the chance that you’ve picked the car is 1 in 3. That means there’s a 2 in 3 chance that the car is behind one of the other two doors.

Monty then opens one of those doors. But because he knows where the car is, he must open a door with a goat behind it.

That’s the key to the puzzle. He is forced to open a door revealing a goat – he can’t reveal the car. Two times in three your original choice is wrong. In those cases Monty has no choice: one of the other doors leads to the car, so he must open the other one, which leads to a goat. One time in three your original choice is right, in which case both the other doors lead to goats and he can open either.

Your original door doesn’t suddenly become more likely to lead to the car when he opens the other door. It still has the same 1 in 3 chance it had when you chose it.

So you should switch, and give yourself a 2/3 chance of the car!

Look at It Another Way

There are only three possibilities.

If the car is behind Door 1, you switch and lose.

If the car is behind Door 2, Monty opens Door 3. You switch to Door 2 and win.

If the car is behind Door 3, Monty opens Door 2. You switch to Door 3 and win.

So switching wins in two of the three possible cases.

But if you’re still not convinced, let’s make the game a bit bigger.

Pick a Card

Instead of three doors, imagine 52 cards lying face down on a table. One of them is the Ace of Spades.

You choose one card. Now, what is the chance you’ve actually picked the Ace of Spades?

It’s 1 in 52.

Now suppose Monty knows where the Ace of Spades is. He turns over 50 of the other 51 cards and every one of them is not the Ace of Spades.

There are now just two cards left.

Your original card had a 1 in 52 chance of being the Ace of Spades. Monty turned over 50 cards he knows are not the Ace of Spades. That doesn’t change the odds about your original choice. It’s still 1/52. If he had no clue where the Ace of Spades was and still turned over 50 cards blind, and only two cards remained, now we are looking at a genuine 50/50. But he knew where it was all along.

In this case, switching wins 51 times out of 52.

So would you switch now?

I would!