I see it now, Warplayer. And I think I see why...

Let me try a table, Warplayer. I'm going to ignore the difference between the two losers, since it's not relevant (as you noted) and it simplifies the tables:

There are three possibilities:

W - L - L
L - W - L
L - L - W

Let's use your example. I chose first, Chaimwitz chose from the two remaining and you were stuck with the last one. Now, Chaimwitz opens his:

W - Lo - L
L - Wo - L
L - Lo - W

Since we know the winner wasn't revealed, line two is stricken from possibility, leaving us with:

W - Lo - L
L - Lo - W

Whether I switch or not would make no difference in my odds, since there are only two rows. If we include the three possibilities where Warplayer opened his, the result would be more of the same.

Now, the difference between this situation and the Monty-Hall Syndrome is that Monty removes _one_ of the losers. He can only remove one. So let's look at the table that way:

Here's the three possibilities:
W - L - L
L - W - L
L - L - W

Now Monty takes away one of the losers from the second or third choices and we are left with:

W - L
L - W
L - W

It isn't a possibility that Monty could reveal both of the losers in the first row. If Chaimwitz and Warplayer have those boxes, it is a possibility that either of them could reveal. But Monty can only open one of them. Half the time he could open the first loser, half the time the second... but _all_ the time he can open only one.

So, switching _is_ only preferable if the person removing the loser knows which is the winner. KelMasterP, I'm not sure if that's what you were saying, but I'm sorry for being so stubborn none-the-less.

OK. Ready for my next lesson, Warplayer...

Based on the puzzle situation, one box is opened. In my program, the "dummy" box is the one that is opened - it is "dummy" because it's neither "my" box, nor "the other guys". By the situation, we know that "dummy" isn't the right one - it represents the box that was opened BEFORE the switch :)

Whoops, missed the edit limit. I didn't see that second paragraph :)

The first run was indeed more accurate of the problem itself. The second program was to show that you get different "results" when the "other guy" does NOT know which booster it's in - ie, when he does know, the outcome is different than if he does NOT know :)

To be honest, I was just as skeptical at first. It wasn't until I wrote the program and SAW the results that I started realizing that it works that way, and with that, I could start trying to discover WHY :)

This thread would be a good example to show to people who believe that "gamers are morons. Sitting around playing with plastic army men". There have been numerous arguments, pro and con and explanations that were well thought and enunciated in easily understood verbage. A fine example of reasonable discussion and interaction.

Thanks DBlizzard for starting this thread and everyone who contributed.

(I would still keep my first pick though if he offered to trade. Now if he didn't make the offer I might worry.):D

I figured out another way to say it that makes it clearer (at least to me). It hinges on the possibility where I am already a winner, but don't know it yet:

W L L

Monty can only open one of them. If Monty opens the first loser, it is no longer a possibility that he could have opened the second loser. That removes opening the second loser from the list of possibilities. If he opens the second loser, it removes the possibility of him having opened the first loser. So, while there are two possible outcomes, it can really only go one way or the other in probability. Only one of the two losers can have been opened when all is said and done. Monty's choice eliminates the other possibility.

If a loser is opened randomly, it's just as likely to be the first one as the second one. But, if a loser is opened specifically, it eliminates the possibility of having opened the other loser.

So there's only a 1 out of 3 chance that I have the winner but don't know it if Monty picks; but a 2 out of 3 chance that I have the winner if a loser is opened randomly.

Did that make sense? I think it did... :p

BTW, is this what is meant by 'restricted choice'?

When you didn't pick the box with the prize, Monty's choice of which box to show you is restricted to the box without the prize. That's an important part of the principle.

Anyone without any knowledge of trick taking card games like bridge, spades, etc. can ignore the following commentary.

The bridge application (where you'll hear this most of the time) deals with having a suit with ace, king & ten on the board (the dummy hand). If the declarer plays the ace and the person playing after the dummy drops either the queen or jack, the odds are that it's a singleton and you should finesse the ten.

In this instance the reasoning is that if he had both the queen and jack doubleton, he should play either randomly. However, if he just had the queen (or just the jack) his choice is resticted to playing the card he has.

A good analysis of the monty hall principle and the bridge application can be found at http://www.acbl-district13.org/artic003.htm

Wow, my wife and I play Pinochle regularly with friends of ours. For Christmass, we are getting them Bridge rules and a good deck of cards.

Thank you Sigma for the perfectly timed note. And for the programming. It is often easier to find why something is true once you know it is true.

A big thank you to everyone who participated in the discussion, even if only to laugh. I love math, but I seldom find people this interested in math puzzles anymore. (I haven't taught math for a few years now, it all Science and Web Design.) I used to have a blast offering bonus points based on puzzles like the one DBlizzard posed here.

And a special thank you to DBlizzard for starting it all.

:)

Well, as a Computer Science major, I do have a little bit of Math background, so I actually do enjoy math. It's not my strongest point, but I do like it :)

However, when it comes to puzzles, I prefer "applied math", such as here. Ie, for me, theoretical math is fine and dandy, but it doesn't strike me as something I want to do unless I can apply it somewhere :)

Switch there could be another horseman in the other booster.