Let's make a deal

Back in the 1970s there was a game show called Let's Make a Deal, hosted by Monty Hall, which created quite a stir when statistics got involved.

The game show went like this. There are three doors; behind one of the doors is a brand new car (Or similarly nice prize) and behind the other two is a goat. And no you don't want a goat. The contestent is then asked to pick a door. Monty Hall would then "reveal" one of the two remaining doors to show one of the goats. This left two unopened doors.

The challenge now is, "would you like to stay with the door you first chose, or switch to the other unopened door". The statistics question is; what is the probability that the other unopened door has the car behind it?

Initially people thought that it was a simple 50:50 bet. There are two doors, one has a car, one does not. But this is why we have statistics, to go beyond common sense and discover strange underlying probabilities. The real controversy started when Marilyn vos Savant's "Ask Marilyn" column in Parade magazine in 1990. She stated that one should always switch. Numerous readers including PhD's and respected statisticians wrote in to her "switch tactic was wrong. Here's why she is right.


Let's say you pick door number one. The probability the car is behind it is then 1/3, since one door has a car, and two have goats. Therefore the probability that the other two doors (2 or 3) contain the car is 2/3. When a goat is revealed to be behind door number 2, that probability does not change. Thee is still a 2/3 probability that the car is behind door number 2 or 3. Obviously don't pick 2, so the probability of door number 3 having the car is 2/3.

Seen another way. All possible setups of doors are  (C=Car, G=Goat)

CGG 

GCG

GGC 

If we pick the first door, then Monty would open these doors

So if we switch our door in the first setup, we loose. If we change on the second we win, and the third we win. Therefore we wind 2/3 of the time. If you don't believe me here's a link to an applet, http://math.ucsd.edu/~crypto/Monty/monty.html. You can play the game until your hearts content, and as you'll notice, you'll win 2/3s of the time if you switch your door. Ain't Stats fun.