In the last issue of The Tottenham Report, I discussed how little many of us understand numbers and to demonstrate this, I posed a probability problem and promised to explain the counterintuitive answer in this issue.
To recap. You are on the TV game show, “Let’s Make A Deal.” You’re given three doors to choose from. Behind two of the doors are goats and behind one of the doors is a car. Whatever door you select, you keep what is behind it.
You make your selection and the host, Monty Hall, opens one of the other doors and shows you a goat. He asks, “Do you want to change the door you have chosen?”
You have two doors left, one with a goat and one with a car. Assuming you want to win the car and not the goat, do you make the change? Do you stay with your first choice? Does it make a difference?
The answer is, you should change; selecting the other unopened door doubles your chances of winning! And here is why.
There are three way to arrange the two goats and the car behind the three doors as shown in the illustration below. In the following example, I always pick Door 1. If I change, I will win two out of three times. If I do not change, I will win one out of three times.
Why is this?
When I choose one of the doors, I have a one in three chance of selecting the door with the car behind it. In other words, there is a two-in-three chance that the car is behind one of the two doors I did not select.
When Monty opens one of the other doors and shows me a goat, there is still a two-in-three chance that the car is behind the doors I have not chosen. Monty has shown me a goat behind one of them, which means there is a two-in-three chance that the car is behind the unopened door I have not chosen.
Counterintuitive, but true.
