© Peter Sanders MMXIV

There is a famous puzzle that goes like this:
You are a contestant on a tv game show and there is a big prize. All you have to do, is guess which one of 20 doors it lies behind.
The doors are numbered 1 to 20, and you pick one (let’s say it is number 1).
The host of the show then gives you some help - he opens one of the other doors to show it is empty.
Then you have a choice - to keep your chosen door or to change your mind. What should you do?

At first thought, the answer is obvious - keep your choice, it’s just as likely to be right as any other door, so there is no point in changing.
But, that’s wrong. You must change your choice...
Why? well, it’s hard to understand with 20 doors, so reduce it to 3.
Your choice is random, so you have ^{1}/_{3} chance of being right and ^{2}/_{3} chance of being wrong. However, the host removes a wrong door.
You now have 2 doors.
Your choice is still ^{1}/_{3} for being right, meaning the other door is ^{2}/_{3} for being right. In other words, the other door is a better choice.
With 20 doors, changing your mind does not increase your chances by much, but it does some and that’s all that is important.