Cori Pringle - July 2000

A Controversial Probability:

Monty's Dilemma

Would you like to make a deal? What? Ok, ok...Years ago there was a game show called Let's Make a Deal. A man by the name of Monty Hall hosted the show. Contestants from the audience had to produce random items from their bags and pockets in order to participate. For example, Monty would ask for a toothbrush. The first person to display one would have the opportunity to win a prize.

There were usually three doors, curtains or something similar to make a choice from, and something good would be behind one of them, say a Mercedes. The other two would generally not be so great; perhaps a couple of AM radios. So the problem of Monty's dilemma emerged. After one makes a choice, Monty will show one of the 'bad' doors that has not been selected. So, is it better to stick with the first choice or switch to the other door?

As most people consider this question, it seems quite obvious that it would not make a difference if one switched their choice. After all, the probability of winning the Mercedes is 1/3 for any door, right? Not so simple...

So, after the contestant makes her or his selection, he or she has a 1/3 chance of having chosen the correct door. But, there is a 2/3 chance that the Mercedes was behind one of the other doors. Monty shows a 'bad' AM radio. So the other door has a 2/3 chance of being your dream car since one of the choices in the 2/3 category was eliminated.

Are you not convinced? I suppose the best thing to do now would be to test it out for yourself. Click here or click here for a couple of simulations to help you see. I would suggest at least 100 trials.

Have you reconsidered? Surely you see that about 2/3 of the time you will win the good prize if you switch from your original choice.

This problem has caused much controversy, but is a fantastic example of a 'things aren't always as they seem' type of problem. If you do not have access to computers and would still like to use a simulation using technology, click here for a program on a TI-82 calculator.