The solution

[yendi]

Warning: This is the solution as presented by a liberal-arts person who hasn't taken a math class in over twelve years. So I may have left gaps unfilled.



Okay, let's go over the one fact that's not likely to be disputed:

After picking your initial box (which we'll call Box A), you have a 1/3 (or roughly 33.3%) chance of having picked the $5000.

That means, of course, that there is a 2/3 or ~67% chance of that money being in one of Box B or Box C.

Now, remember (as trochee emphasized), the Prizemaster knows where the money is.

He removes a box (we'll call it Box B) that was known not to contain any money, and never did.

Box A still retains a 33% chance of being the correct box. That's because it was picked before the state of Box B was known.

But remember -- the chances of the money being in one of the other two boxes were 2/3. Box B is out of the way. That means that the odds for the money being in Box C (which is, of course, "one of Box B or Box C" remain 2/3.

Find this troubling?

The argument that most folks have with this is best articulated by force_of_will in this comment and the follow up. And had the Prizemaster been choosing whether to remove a box, or if the PM was choosing randomly and happened to pick an empty one, this logic would be correct.

Let's pretend there are 100 boxes. You pick one. Then the Prizemaster, knowing where the money is, removes 98 of the remaining boxes which contain no money. He again offers you the chance to switch. Still think the box in your hand, at 1%, is the way to go?

(that last example is swiped from Raymond Smullyan, btw)

Comments

if I haven't done this already...

[dopple.livejournal.com]

You have been judged significantly geeky to join de (http://www.livejournal.com/userinfo.bml?user=multi_genre_fan)Multi-Genre Fandom (http://www.livejournal.com/community/multi_genre_fan/). Come on over and bask in de dorky afterglow!

Re: if I haven't done this already...

[yendi]

Woohoo! Dorky afterglow!

[mfree.livejournal.com]

If the prizemaster has knowingly removed boxes containing no money, and you know now of two boxes, one that you are holding and one that is on the table, and either could contain money, how is it that the odds are not .5?

[yendi]

Because the prizemaster removes the empty boxes after you've made your initial 1/3 (or 1/100) choice.

[trochee.livejournal.com]

The Smullyan example doesn't do it for ya?

[photognome.livejournal.com]

(polishes fingernails) its hard work being this right all the time...... ;-)

[terracinque.livejournal.com]

Oh right, right! I remembered wrong. What you have above is what my friend and I demonstrated.

[beowabbit]

The little piece of information that’s important is that it’s not random that the prizemaster is picking an empty box. If the prizemaster picked one of the remaining two boxes at random (with a 1:3 chance of revealing the prize), then everybody’s first intuitions about this problem would be correct.

[robyn-ma.livejournal.com]

Or there's my solution:

You hit the Prizemaster in the head with a broom, open all the boxes, take the money, and run like hell. :)

[yendi]

Thinking outside the box there. Or inside the boxes, as the case might be.

[force-of-will.livejournal.com]

It "works" that way as long as you play from the point of view of the "player". The Prizemaster has a whole different problem. Then there is the perspective outside of the game...

If you believe that you get to "keep" that 1% chance, you should choose to switch boxes. If you believe that you get a whole new problem, it doesn't matter to switch. For the Prizemasters part, he knows that you'll never get to keep the 1% chance, you'll wind up having to choose from one of two boxes with everything that came before being thrown out.

I don't believe that you get to keep the 1% chance packed up in the box. It was had, is gone, and you were never going to get that particular shot to win the prize because the Prizemaster throws it out. You were always going to get a choice that he gives you in the end, one of two boxes.

The problem reminds me of those sorts of riddles that give you a lot of extraneous things only to confuse you with them.

Will

[trochee.livejournal.com]

I really don't understand what you mean by "keep" a chance.

[brujah.livejournal.com]

My. Head. Just. Exploded. (thud)

[yendi]

Cleanup on Aisle 4!

[clawfoot.livejournal.com]

My brain hurts.

[jet-li-wannabe.livejournal.com]

I usually explain it to my students this way:

Scenario 1: I (the player) never switch.

In this case, there is clearly a 1/3 chance of me winning.

Scenario 2: I (the Player) always switch after the Prizemaster makes his move.

If my initial choice is correct (1/3 chance) I lose.

If my initial choice is wrong (2/3 chance), then the box that the prizemaster didn't open has the money, so when I switch, I win.


Conclusion:

If I never switch - I win 1/3 of the time.

If I always switch - I win 2/3 of the time.



The funny part is that if I always switch, then from the perspective of my initial choice, if I'm right, I lose and if I'm wrong, I win.

[yendi]

That explanation makes sense -- keeps it all in the context of the first choice.

[rubian77.livejournal.com]

Ok, but did I win the money or not?

[mfree.livejournal.com]

*waves hand* This is not the money you are waiting for...

[unwilly.livejournal.com]

I think some people are confused, as I was, between percentage of winning and types of outcomes.

You have a 2/3 chance of winning if you switch after the empty box is removed, but there are still only two outcomes win/lose, which makes it look like a 50% chance that either box has the money.

[yendi]

*nod* Makes sense. I hadn't actually realized it, but that's a very logical and likely source for the fallacy.

[wundergeek.livejournal.com]

Thank you for reminding me why I'm an *art* major. Good Lord. I just spent twenty minutes reading all of the comments in this thread and I'm still not getting why switching boxes is a good thing. I mean - if the one he removes is empty, why doesn't it stop counting?

I know, I know, there are tons of ways to explain it. I read all of the links and it just doesn't make sense to me. I guess you have to be a linear thinker to get it. Maybe.

[trochee.livejournal.com]

I'm surprised nobody's posted the wikipedia link yet: Monty Hall problem (http://en.wikipedia.org/wiki/Monty_Hall_problem).

[trochee.livejournal.com]

And somebody's done an Empirical solution (http://en.wikipedia.org/wiki/Empirical_solution_of_the_Monty_Hall_problem) with a perl script that executes 3000 iterations of each strategy.

great minds think alike ...

[fmi-agent.livejournal.com]

... at least when coming up with plausible explanations :) i discussed this with my LJ-friends in '03 ... some very similar arguments came up. (my take on the story here (http://www.livejournal.com/users/pbmath/255625.html))