Wombat School

The answer is no, it is not to your advantage to switch. At first there were three unknowns, and now there are two, which leaves a 50-50 chance of getting the car regardless of which box you choose. Using C for the car, and G for the goats, a simple truth table shows this:

       box1  box2  box3
case1   G     G     C
case2   G     C     G
case3   C     G     G

Let's take the cases one by one with the following premises: you always pick box1 and you'll always take Monty's offer to switch. The following cases detail the outcome depending on which box Monty reveals.

So, in case1, if Monty exposes box2 and you switch to box3, you win.
In case2, Monty won't expose box2 (because it contains a car).
In case3, if Monty exposes box2 and you switch to box3, you lose.

Now, in case1, Monty won't expose box3 (because it contains a car).
In case2, if Monty exposes box3 and you switch to box2, you win.
In case3, if Monty exposes box3 and you switch to box2, you lose.

The experiment can be repeated as necessary with you choosing box2 and then box3 or with you opting not to switch, but the results will be the same each time.

If Monty shows box2 and you switch, then there's one chance for a win and one chance for a loss. If Monty shows you box3 and you switch, there is again one chance for a win and one chance for a loss. Two wins and two losses make a 50-50 chance of getting the car, therefore it doesn't affect your odds if you switch.

What's that? Not satisfied? Try the Katydid school!

The car-goat-goat Problem / corby@intuit.com
December 1994 (updated April 6, 1998)