# The Monty Hall 'problem' is simple

13 September 2013

A long article on the BBC website about the ‘Monty Hall’ problem - a favourite mathematical brain-teaser - once again suggests you need to be a mathematical genius to understand it:

Monty Hall problem: The probability puzzle that makes your head melt

(Monty Hall is a game show host.) Basically, say you have three boxes - two containing goats and one containing a Caddilac. You pick one of the boxes (but don’t open it, so you don’t know what’s in it). Monty opens the one of the other boxes which he knows contains a goat, and asks you would you like to stick with the box you have or choose the other one.

What are your chances of winning the Cadillac if you decide to switch boxes? Most people say 50-50, but actually it’s 2 out of 3 in favour if you switch. The reason for this is what causes people to go into a spin.

However the problem is deliberately made difficult by leading you to think of both goats as the same. If one goat is black and one is white, then you have three possible paths.

1. If you picked the white goat, Monty will reveal the black goat and the other box contains the Caddilac.

2. If you picked the black goat, Monty will reveal the white goat and the other box contains the Caddilac.

3. If you picked the Caddilac, Monty will reveal one of the goats, and the other box contains the other goat.

So choosing to switch gives you two chances out of three of getting the Caddilac.

Of course in reality it's meaningless for one attempt. If you make 30 attempts and always choose to change, you would win 20 times or thereabouts. But if you only have one go then though you are more likely to get the cadillac by changing, that will be no consolation if you land up with the goat (either of them).

Monty Hall problem: The probability puzzle that makes your head melt

(Monty Hall is a game show host.) Basically, say you have three boxes - two containing goats and one containing a Caddilac. You pick one of the boxes (but don’t open it, so you don’t know what’s in it). Monty opens the one of the other boxes which he knows contains a goat, and asks you would you like to stick with the box you have or choose the other one.

What are your chances of winning the Cadillac if you decide to switch boxes? Most people say 50-50, but actually it’s 2 out of 3 in favour if you switch. The reason for this is what causes people to go into a spin.

However the problem is deliberately made difficult by leading you to think of both goats as the same. If one goat is black and one is white, then you have three possible paths.

1. If you picked the white goat, Monty will reveal the black goat and the other box contains the Caddilac.

2. If you picked the black goat, Monty will reveal the white goat and the other box contains the Caddilac.

3. If you picked the Caddilac, Monty will reveal one of the goats, and the other box contains the other goat.

So choosing to switch gives you two chances out of three of getting the Caddilac.

Of course in reality it's meaningless for one attempt. If you make 30 attempts and always choose to change, you would win 20 times or thereabouts. But if you only have one go then though you are more likely to get the cadillac by changing, that will be no consolation if you land up with the goat (either of them).