The Monty Hall Problem is a counter-intuitive brain teaser based on the TV game show “Let’s Make a Deal” hosted by Monty Hall.  In the game, you are given 3 doors.  Behind one door is a car and the other two doors have goats.  Only the Host knows what’s behind the doors and once you have selected your door, he opens one of the other doors with a goat and offers you the chance to choose a different door.  Should you switch your choice or just stick with your original decision?

The answer is that you should switch. It was proven mathematically back in 1975 to provide a 2/3 change of winning the car if you switch but only a 1/3 change if you keep your original, but why are the odds in your favor when switching?  The reason is that they are removing one of the wrong choices for you.

When you first choose, you had a 1/3 change of picking the correct door.  Whereas there is a 2/3 chance that one of the other doors had the Car.  Now imagine that your initial pick was wrong.  That means that the car must be in one of the other two doors.  If they open the door that it is NOT in, then the car must always be in the other door.  So… in other words, switching effectively gives you the better of two doors which means that 2/3 of the time you will win if you switch.

The illustrations below are an excel file that I generated with random numbers, so it does not add up to exactly 1/3 in each column but still illustrates the point.  If you look at the middle illustration it shows you what would happen, if you picked door 1 as your first choice 30 times.  You will notice that if you picked a goat on your first pick, Monty eliminates the other goat every time.  Consequently, if your first guess was wrong you have a 100% chance of picking the right door the second time, which represents 2 out of 3 doors.  To state that again, because your odds of picking the right door the first time were 1/3 and if you were wrong you are now guaranteed to get it right switching.  this means that staying with the same door gives you a 1/3 chance of being right and a 2/3 change if you switch.

I was nice of Mr. Hall to give you the best of 2 doors effectively combining their odds to give you a 2/3 chance.

Random Doors

If You Pick Door 1

Final Choice