Monty Hall

[andrewducker]

[Poll #1770413]

Explanation

I have known what the answer was for ages, but for some reason it only "clicked" in my head today. You can blame sarahs_muse for triggering it.

Comments

[marrog.livejournal.com]

Cannot believe no one told me the million doors version when trying to explain it to me. It would have taken me way less time to 'get' it if they had.

[steer.livejournal.com]

Yes... the million doors version makes it obvious doesn't it.

I confess I got it wrong when I first encountered it but I had had four or five pints.

[andrewducker]

It does seem to be exposing something in the human brain that doesn't quite match up to reality. Why my brain could see it properly for a million doors, but not for three doors is a mystery to me.

[naath.livejournal.com]

I first heard of this one when I attend the RI Christmas lectures given by Ian Stewart; my Mother (a maths teacher) didn't believe his proof (which had been framed for the, fairly young but keen, audience) although I did.

It's very irritating to explain to people in pubs, much easier with pen and paper :-p

[bracknellexile.livejournal.com]

I actually find it easier to explain in pubs cos you can use upturned empty glasses and a penny to illustrate it. Then people can see where the prize is and so they can see why they should switch - much like the diagram in the "decision tree" section of Andy's explanation link above.

[innerbrat]

Mind you, 1/3 is a really high probability anyway. You'll get the goat if you switch enough times that you'll be convinced it's random, because you're human

[channelpenguin.livejournal.com]

though I got it right - it was for 2 wrong reasons

1. [instant thought] It wouldn't be a 'problem' if the instantly obvious answer (doesn't matter) were true.
2. [immediately following thought] My brain rephrased it as a 1 in 2 (50% chance) once you have 2 doors left and know one of them is a goat and one is a car

I failed yesterday to satisfactorily explain to Steven the 'correct' (66%) ratio, so I'd be interested in this "million doors" explanation.....? Link?

I must ask my sister this one ans see what she says - she is numerically ultra-sharp and a statistician (insurance risk analyst) by trade (well at least til she went management...)

[andrewducker]

The link is part of the wikipedia article in the post.

For how I internalised it, see my response to dpolicar below.

[bracknellexile.livejournal.com]

Shouldn't the poll really have the "It doesn't make any difference if I switch or not" option, given that's the answer most folks give to the MHP?

[theweaselking.livejournal.com]

That would be "I do not believe that switching is beneficial"

[dpolicar]

Neither answer quite covers it.

I "believe" that switching is beneficial because of a combination of peer pressure, having found some argument that seems compelling that suggests it, and having sat down for an hour with a deck of cards and run Monty Hall simulations and counted the results.

But it has never "clicked" for me, which makes it a very unstable belief.

Then again, I have a lot of beliefs like that.

[dpolicar]

I should add that I also "believe" it on the philosophical grounds that probabilities (at least macroscopic linear ones) are quantifications of ignorance, not facts about the world, so it ought not surprise me that manipulating my ignorance will make things more or less probable.

But really internalizing that is hard.

[andrewducker]

I wrote some code this afternoon to prove it to myself.

What I discovered was that either I had chosen the prize with my initial guess, or I had not. Opening doors that I hadn't chosen did not change the correctness of my initial guess. And therefore, there was always a one in three chance that my original guess was correct.

Therefore, as there was only one other door to choose, that door must have a two in three chance of being correct.

[theweaselking.livejournal.com]

I don't like the answers because "I believe" is not correct. I *know* and can *trivially prove* that switching is beneficial. There is no belief required.

[meaningrequired.livejournal.com]

The book I told you which had the solution? Recommended you should always change. I voted you shouldn't because I was all for going with my book's answer, but I remembered wrong. I did remember it wasn't what you expected, so I double bluffed myself!

Although, it still feels uncomfortable, and anyways, its a travesty against goats...

[a-pawson.livejournal.com]

Exactly - why won't anyone think of the goats?

[bracknellexile.livejournal.com]

I've heard the million doors explanation before and yet people still go, "Ah, I get it for a million doors and the odds are 999,999 to 1 but if there are only three doors, doesn't it come back to 50/50?" because they can't make the leap from a million to three. I always tried to explain it with just the three to avoid confusion.

I also wonder if people don't get it because the choice seems to be between "car" or "goat" and there's an inherent 50/50 feel to that. If the three doors hide a car, a goat and a sheep, then (unless you're of a particular wool-loving persuasion) there are now three different outcomes rather than just cars and goats.

Some explanations assume you pick door 1 with either a goat or a car behind it. In my head that also adds another level of complexity because, if you don't spot the symmetry in the problem then the next question is: "But what if I start with door 2?"

By differentiating between sheep and goat, you can also do away with the door numbers in explaining it and keep it simple:

It then comes down to:
If you picked the goat first, the host reveals the sheep. If you switch you WIN.
If you picked the sheep first, the host reveals the goat. If you switch you WIN.
If you picked the car first, the host reveals one of the animals. If you switch you LOSE.

You don't know which one you picked but you will always be in one of the three scenarios above. In two thirds of the scenarios switching wins you the car, in the other one it doesn't, ergo switching is good 2/3 of the time. Job done.

Maybe it's just me but that seems simple and straightforward in my head. Simpler than trying to explain why "If you picked a goat then the host reveals the other goat" has to be counted twice for the two combinations of goats.

[theweaselking.livejournal.com]

The explanation I always like to use is that you're not picking between your first choice and the one remaining door. You're picking between your first choice and *the best of all the other possible choices*.

[cheekbones3.livejournal.com]

As a statistician, I couldn't really afford to get this wrong...

[theweaselking.livejournal.com]

PS:

You run Jurassic Park 3.0. Your lead scientist tells you he has just bred two new Velociraptor babies. You fire him immediately for breeding Velociraptors, but now you have a problem: Those are billion-dollar killing machines, so you can't afford to just write them off if it's possible you can keep them.

If the new babies are female, you can safely put them in your tank and they can't breed, because you only keep female Velociraptors. If they're male, you'll have to eat the loss and sell them to the Roast Dino Hut franchise, or something.

You ask the newly-promoted Head Scientist if the babies are female. "Yes," she says, "I've checked the first one, and it's female!".

What are the odds that, when she checks, the second one will also be female?

(assume each baby has a 50/50 chance of being female)

[andrewducker]

50%.

There are four possible worlds I am living in:
MM
MF
FM
FF
Because she has proven that I do not live in either of the first two worlds, I am left with two possible worlds, one with a male second raptor, one with a female. And thus a 50% chance that I am sharing it with two female raptors.

Do I win a goat to feed to my raptors?

[ptc24.livejournal.com]

Consider the Ambiguous Monty Hall problem.

Somehow, you end up being one of the first contestants on this new game show. The host, who you have learned is called Monty Hall, presents you three doors, tells you there's a prize behind one, and nothing behind the other two, and asks you to pick a door. You pick a door, expecting him to open it. But instead, he opens another door, and asks you if you want to change your mind. You did not expect this. You know no more than what you have been told.

Unanswerable questions:

a) Does he always do this?
b) Does he know where the prize is?
c) Is he on your side or not?

Calculating the probabilities requires an answer to these questions.

[doubtingmichael.livejournal.com]

Good point. I believe this was the case in the original, real-world Monty Hall quiz show. Sometimes he would open a door, sometimes he would take your first pick. At this point it is probably rational to go:

(a) The man likes to save his show's budget.
(b) He does this sort of thing for a living.
(c) He is probably better at mind games than you.
(d) I should stick with my first choice, because that way I guarantee my one-third chance of being right.

Exception: if you were an utterly cute thirteen year old, swap if he suggests it. Because if he's nasty to you, his sponsors will be cross with him.

[rhythmaning.livejournal.com]

I don't believe switching is important because you can still choose the original box and hence have 50% chance.

[andrewducker]

Sadly not.

You had a 1 in 3 chance when you picked the original door. Opening doors on the other side doesn't change that chance.

If we could take you, once the other door has been opened, knock you out, remove all of your memories, and then ask you to choose again with no bias, _then_ you'd have a 50:50 chance of getting it right. But as you already know things, this isn't the case.