This problem is known commonly as The Monty Hall problem (after the presenter of a US game show) but it was initially found in
Bertrand's Calcul des probabilites (pub.1889).
On the game show theme, imagine you have three closed doors in front of you. The completely honest and truthful game show host
tells you that there is a million dollars behind one of the doors but just 1 dollar behind each of the other two doors. The
host is aware of which amount of money is behind which door.
You are to pick a door, the host will then open one of the other doors to reveal 1 dollar. He then says you can switch your
choice to the other unopened door.
The question is should you do it? Are you more likely to win the million dollars if you stay with your original choice or choose
the other door?
The answer is, you must always switch because there is twice as much chance of you winning the million dollars than if you
stay with your original choice.
Before you all mutter in disbelief, here is the reasoning.
As you do not know which of the two unopened doors has the million dollars behind it, it would seem that each door is just as
likely to be the one to make you a millionaire. Actually, because the host knows which one has the million, he will always
know which one to open to reveal the 1 dollar. The door that you have chosen will have a 1 in 3 chance of being the winning door
as you had a choice of 1 door out of 3. However, as the host has opened a losing door, it gives the final unopened door a 2 in 3
chance of being the million dollar door.
Therefore, 2 in 3 chance is better than the 1 in 3 chance you have of the door you originally chose.
So if you find yourself in that situation, always switch and you may win yourself a million dollars.
