Recently I had some fun by testing some “smart” friends on the classic Monty Hall problem [1]. They performed as expected – they were flummoxed and in denial. This again demonstrates that probability theory does not come naturally to us and if we don’t have a good understanding of probability, we stand to leave value on the table.

The Monty Hall problem: 3 closed doors. A car behind one of them, and goats behind the other two. Contestant picks a door. The host, Monty Hall, knows where the car is. He opens a door that they have not picked to reveal a goat. Contestant is offered the choice to switch. Would you switch? Most people believe the odds are 50/50 between their original choice and the remaining unopened door. But, they’re wrong! It is better to switch. Why?

The solution: the probability that your first choice was correct is 1/3. Since the car is behind a door, the probability it is behind one of the other two doors is 2/3. Now, when Monty opens a door to reveal a goat, that 2/3 probability that applied for two doors, is still 2/3, but now applies only for the remaining door. So, your first choice gives a 1/3 chance, and switching gives you a 2/3 chance.

Still not convinced? Imagine there are 10 doors – a car behind one, goats behind the other nine. You pick a door and hence have a 1/10 chance, with 9/10 probability the car is behind the remaining 9 doors. Monty opens 8 doors to reveal 8 goats and leaves your door and one other unopened. That 9/10 probability for the 9 other doors now applies only to the door that he hasn’t opened. So, switching will result in you winning the car 9 times more often than staying with your first choice.

Clearly this is an artificial problem, but it does illustrate how misleading our intuition can be when it comes to probabilities and the implications of making a poor strategic choice in an environment of uncertainty.

For some amusement, the original responses by some very smart people are a must-read [2]. They include “As a professional mathematician, I’m very concerned with the general public’s lack of mathematical skills. Please help by confessing your error and in the future being more careful.” and “You blew it, and you blew it big!”

References:

[1] https://en.wikipedia.org/wiki/Monty_Hall_problem

[2] http://marilynvossavant.com/game-show-problem/