Statistical Paradox: The Monty Hall problem.

Probability has a pretty common usage in my field (Psychology). It’s great, but some aspects are pretty confusing. My favourite example, is the “Monty Hall” problem. Loosely based on an American game-show, and given popularity in a column by vos Savant, it’s a really good example of what is called a “Viridic Paradox”, a paradox where the answer is the opposite of what you’d expect.

 

The paradox asks a simple question, but to fully understand what that question is I’m going to have to play a quick game with you.

Say I’ve invented a new (albeit low budget) gameshow. The idea is that on a table in front of you I’ve placed three small boxes; Beneath one box is a £1 coin; underneath the others is a penny. You don’t know which box contains what, and the idea of the game is to choose the one with the £1 coin in

It’s your turn first, and you get to pick one of the boxes (box A), but you don’t get to see what’s in it just yet. After you’ve made your choice, I’ll eliminate one of the boxes that I know contains a penny, so that there are now just two boxes on the table, and I’ll give you a choice “Would you like to change your decision and choose the other box, or stick with the one you originally chose?”

Now, here is the question at the centre of the paradox: Is it to your advantage to switch to the remaining box?

The answer is yes! By switching to the other box (the one you didn’t initially select) you increase the odds of getting the £1. In fact, the odds that the £1 is in the other remaining box is 66%. 

This is really counter intuitive (which is why its an example of a Viridic paradox). It seems obviously wrong, because there are two boxes so the odds should be 50:50. A good way of explaining how this is correct is to increase the number of boxes (dramatically).

Let’s say we play the same game again, except with a million boxes (I don’t even have that many pennies. Sad.). I give you the same option, you get to choose one box but can’t see whats inside. Same as last time, I go and remove box that I know contains a penny. The difference in this game, is that I also remove 999,998 other boxes that all contain pennies and ask you the same question “Do you want to switch to the other remaining box”.

You had a 1:1,000,000 chance of correctly choosing the £1 the first time, and now that I’ve kindly gone and removed every box, it seems like the best decision is to switch to the only remaining cup: It’s far more likely to contain the £1 than your original choice.

Revert back to the first game now. The chance that you chose correctly is 1:3, I’m not changing those odds at all, but by eliminating one of the wrong choices I’ve increased the odds of the other box containing the £1 up to 2:3. This is why its more likely you’ll get the £1 by switching.

Next time you are looking at “Deal or No Deal” I fully expect you to demand that person switches their box when given the chance.

 

 

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