Variable Change: The Monty Hall Question and the Movie 21

My wife Kristin and I recently enjoyed watching the movie, 21. In the movie, loosely based on a true story, a professor recruits his brightest math students into a card counting scheme. They win millions of dollars playing blackjack.

In one scene, the professor poses the following question to his class – the classic “Monty Hall” question: There are three doors. Behind one door is a new car. Behind the other two, goats.

He asks the bright young student (whom he is about to recruit into the blackjack group) to pick a door. The student picks door number one.

The professor then opens door number three to reveal a goat. At this point, he gives the student a choice: to stay with his original choice, or to switch to door number two.

The student chooses to switch to door number two.

The professor then poses the question of why. Many people would stay with their first choice, but why?

The  young student then explains that by switching, he has a 66% probability of choosing the car.

I couldn’t figure this out. It seemed to me that it would be a 50% chance… I mean now he has to make a choice and either it is the goat or the car…

Here is the solution – it is really quite simple, but not immediately obvious.

1. When he makes his first choice he has ONE chance in THREE of making the correct choice (although it isn’t immediately revealed to him).

2. After Monty opens the first goat door (we know Monty knows where the car is) we are left with a choice of two doors.

3. The correct choice now depends on whether we were right, or wrong with our first choice. If we were RIGHT on our first choice, switching doors will LOSE. If we were WRONG on our first choice, switching doors will WIN. Since we only had one chance in three of being RIGHT the first time, and TWO chances of being wrong, the odds are that we are WRONG and therefore switching gives us a TWO in THREE chance of being correct if we SWITCH.

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