A Door Puzzle

Let's take some liberty here, and assume we can post both puzzles and jokes here. Here is a puzzle.

You are a contestant in a game show with three doors. Behind one door is a national winning model, and behind the other two doors is a piece of solder. You choose a door. The MC then opens one of the other doors, and shows you a piece of solder. He then gives you a chance to switch your choice to the other unopened door, or stick with your original choice.

What should you do? Should you stick with your original choice, or should you switch to the other door?

Well, Larry, was this a random choice or a preconceived choice? Usually on game shows they know which door is which. So, whichever door you pick originally, they are then going to show you a door that isn't. That being the case, nothing has changed. Your odds originally were one in three. They remain that way between either of the two remaining doors. (Nothing changed since the EmCee is going to pick the "solder" door under any circumstances.)

However, if this was a "random" choice, then the door you did not pick is a 50-50 shot of being solder or a model (airplane, I assume, not a girl. Wouldn't want that.) But, then, after going through all this, and having now eliminated one door, that now makes your original pick a 50-50 shot as well. So, the odds remain the same. Change if you want. Don't change. Shouldn't make any difference. (If your door had been interchanged with the door that was not picked before the EmCee made his choice, the odds would remain the same as they are now when it wasn't in the "group of two". No difference.)

Actually, switching wins two times out of three! Sticking with your original choice ones one time in three.

For the most part, people don't think much of this problem. Of those who do (because they are good logicians, or mathematicians, or statisticians, or some such) follow Leonard's logic.

>. . . So, whichever door
>you pick originally, they are
>then going to show you
>a door that isn't.
>That being the case, nothing
>has changed. Your odds
>originally were one in three.
> They remain that way
>between either of the two
>remaining doors. (Nothing changed
>since the EmCee is going
>to pick the "solder" door
>under any circumstances.)
>
>However, if this was a "random"
>choice, then the door you
>did not pick is a
>50-50 shot of being solder
>or a model (airplane, I
>assume, not a girl.
>Wouldn't want that.) But,
>then, after going through all
>this, and having now eliminated
>one door, that now makes
>your original pick a 50-50
>shot as well. So,
>the odds remain the same.
> Change if you want.
> Don't change. Shouldn't
>make any difference. (If
>your door had been interchanged
>with the door that was
>not picked before the EmCee
>made his choice, the odds
>would remain the same as
>they are now when it
>wasn't in the "group of
>two". No difference.)

The correct answer can be arrived at with Baysean methods. For us lay people, it easier to see the correct answer by imagining you are the MC instead of the contestant. The MC has no doubt as to the correct solution! Imagine standing being behind the doors with full view of all three items.

What the MC sees is that the contestant picks "solder" two times out of three. The MC simply shows the other piece solder. In other words, two times out of three the MC has no choice to make. Two times out of three the model (indeed an airplane, not a girl) is behind the other door. So switching wins two out of three.

When the contestant picks the model first, the MC shows one of the solders. That happens one time in three. Switching, therefore, loses one time in three.

OK, you win. My chances were one in three at the beginning, and remain one in three if I stick with my original pick. However, if I switch after the emcee eliminates one wrong choice (after I have picked) and I then switch, my chances improve to two in three. But think of the fool I make of myself on national television if I change and THEN end up with the solder when I could have stayed with my original choice and won.

Hmm, come to think of it, you never told us what type of model was behind door number three.

HUH? I think what Leonard said was going to be my NEXT guess!