The “Choosing Among Three Doors” Problem

“On a game show, a new car is hidden behind one of three doors. You choose Door 1. The host opens Door 3 — empty — and gives you the option to change your choice. Would you switch to Door 2 or stick with Door 1?”

This question is often called the “Monty Hall Problem”, after the show “Let’s Make a Deal”, and was discussed in the movie “21”.

The question can be answered using knowledge of basic math probabilities.

Probability is always described as a number between zero (no change of something happening) and one (100% chance of something happening), and is calculated using the formula P(outcome you desire) = \frac{desired}{possible}.

When you make your initial choice between the three doors, the chance is \frac{1}{3} that your choice has the prize behind it. Thus, the probability that you have chosen correctly is \frac{1}{3}, and the probability that you have chosen incorrectly is \frac{2}{3}. So, there is a \frac{2}{3} chance that the car is behind either Door 2 or Door 3.

When the host opens Door 3, revealing that it does not contain the price, he has given you additional information, but the original odds remain the same.  There is now a \frac{1}{3} chance that your original choice (Door 1) was correct (Door 1) and a \frac{2}{3} chance that one of the other doors (now, only Door 2) contains the prize, and you should switch to Door 2.

This requires assuming, of course, that the host always offers the chance to switch and that you can’t gain any information from his behavior.

If it’s difficult to see why this makes sense, imagine the same scenario with a million doors.  You pick Door 1, and the host opens 999,998 other doors, showing them to be empty, leaving only your door and one other door.  Would you switch?  There’s a \frac{1}{1,000,000} chance that you picked the correct door, and a \frac{999,999}{1,000,000} chance that the other remaining door holds the prize.

The GMAT Test includes many questions that test your ability to break down and solve problems involving probability.  If you like problems like this, you might enjoy taking the GMAT and earning an MBA.  Contact Bobby Hood Test Prep for information about classes and tutoring, either locally in Austin or online via The Princeton Review’s LiveOnline classrooms!