3 boxes problem ( 3 door game show )

This is a problem that I've seen used in a movie about math students and a professor beating Las Vegas at blackjack.

And if you like Canadian movies, there is a very similar movie called The Last Casino. (I think that the last casino is > 21)

IMDB says Last Casino = 7.2, 21 = 6.7, ( 7.2 > 6.7 ) so I conclude that this is strictly true. If you were in the group of students that Prof Heap assigned a movie to watch, you might want to try one of these or make it a double feature dvd microwave popcorn stay at home and sit on the couch night. Already, I'm digressing.

The problem is described something like this: (Once upon a time)

You are a contestant on a game show where there are 3 doors. (You are always choosing doors in these questions!) They are named (aptly) door#1, door#2, and door#3.

Behind one of the doors is a Prize. But you don't know which door. And there isn't a knight or knave to be found. so you have to guess.

If it is an American game show, then you might win a car.

If it is a Canadian game show then some sort of foldable magazine rack and some Rice a Roni. (The San Francisco Treat?) Actually my Grandfather once won a couch on Definition when I was quite small. This was better than the foldable magazine rack. Too bad he already had a couch. Oh well. I digress again. Let us play super-pretend and imagine that this is an American game show with the better prizes and the gorgeously gesturing spokesmodels.

Ok To recap: there's 3 doors one door has a prize, and the other 2 don't. You have to guess. People in the audience shout numbers at you. You can't phone a friend, and all of your lifelines are used up.

You pick door #2, since it is even (n=2k) and you don't like odd people or numbers. (n=2k+1) (Its just that pesky +1 you just don't like.)

Then the super-suave game show host, says that he'll reveal to you what is behind one of the two doors that you didn't pick. Thanks a lot buddy, you think. Thanks a lot. Way to go making me feel bad. On Cable TV no less. Everyone is watching. I hope it isn't the car. I hope it isn't the car. Then I might still have a chance.

Invariably, the door he shows doesn't have the new car and the trip to Tahiti with the spokesmodel, but instead a mule and oxcart with a hearty load of stale green bacon fat. Now you are given the option to stay with your door #2 or pick the other door that the host didn't open.

Now, if you're an emotional and sentimental person, you'd stick with door #2. (We'll always have Paris!) Since you've valiantly taken CSC 165, you've become a hard and cold logician. With your steely eyes and heart of pure silicon, you coldly calculate your options. Lets see . . . 3 doors. I pick one. Prize has a 1/3 chance of being behind each door, so it doesn't matter which door I choose.

"I'll stick with door #2 Alex. That's my final answer."

Cue the sad music. (Wah Wah Wah.) You just won the wall clock from LM121. (What time is it? twenty after six!) Ah you say, I knew it was rigged. I never win at anything! (I once won a skateboard, but I lost it)

Counter-intuitively, the best strategy in this problem is to ALWAYS switch doors.

If you pick door #1, and The host reveals door #2, then switch to door #3.

More often than not you will win with this strategy, and not because the spokesmodel is quickly driving the car from door #1 to door #3. (That would be cheating)

The correct explanation, as I understand it is to partition the set of doors into 2 sets. Not 3 as you'd imagine.

set you picked = 1/3
set you didn't pick 1/3 + 1/3 = 2/3

When the host opens one of the doors you didn't pick, the probability of the prize being in the set that you didn't pick is still 2/3 and not 1/3.

At first glance this seems unlikely. Impossible even. Try it again, but with a far longer (and more boring) game show where you pick a door, and then the host opens 998/1000 doors and then asks you to stay or switch.

Of course you switch!

in this case your original door is worth 1/1000
and the set you didn't pick is 999/1000.

The trick to remember is that the Host knows the number of the door that the prize is behind. The host will never ever open that door. There would be no drama and both the audience and the contestant would feel cheated by the host. It is much more exciting veiewing if the contestant is forced to "stay" or "switch."