h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Entry: The Monty Hall Problem - A891065
Author: Gnomon - U151503

This is an explanation of a very confusing probability paradox. I hope that after reading it you can understand what is going on.

Recumbentman raised some issues in a conversation attached to the entry itself which I'm going to have to deal with. It will require part of this to be rewritten. I'll try and do that this evening.

I've made some changes and hope that it is now more convincing.

wow - most thought provoking entry I've read so far! Must admit I am equally convinced by all the arguments mentioned in the following conversation though . . .
I don't want to go try it out though, I want to understand it purely mathmatically so to speak, rather than via messy real life data
so I guess the explanation isn't totally convincing yet
it is written really clearly though!

wow - most thought provoking entry I've read so far! Must admit I am equally convinced by all the arguments mentioned in the following conversation though . . .
I don't want to go try it out though, I want to understand it purely mathmatically so to speak, rather than via messy real life data
so I guess the explanation isn't totally convincing yet
it is written really clearly though!

sorry

Let me see if I've got this right:

First choice there's a 2/3 chance of getting an empty door.

which means that

There's a 2/3 chance that Monty will be forced to open the only remaining empty door.

which means that

there's a 2/3 chance that the only door left will have a prize behind it, so, you should switch.

seems simple to me. But then, I'm Canadian, just like Monty.

If you look at the conversations attached to the entry itself, Anhaga, you will see that some people find this very hard to see. That's why it is worth writing an entry about.

I think the entry is fine now, but I already knew the correct solution, so I don't know if it will convince newcomers to the problem.

Hed

I'm still equally convinced by both arguments, and I believe which ever one I've just read. Mathmatical tart.

I like this explanation of a difficult problem (one I have nearly come to blows over!)

The only bit I didn't like was the section 'still not convinced' where you attempt to explain one difficult-to-imagine problem (monty hall) in terms of a another difficult-to-imagine problem (a room with a million doors). 1,000,0000 rooms is simply hard to grasp - and also is such a big number you have to introdces a change in the way the game works...

Here's an alternative paragraph

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Still Unconvinced?

Many people are still unconvinced by this explanation. Try this way: The solution can be made easier to appreciate by increasing the number of doors.

Suppose that there are TEN doors in the studio, as before one of them concealing the prize. Just as before, you pick one.

Now Monty opens EIGHT other doors, all empty, leaving just one door still closed. Just as before he now and invites you to swap: the 'fifty-fifty' option.

But wait! Consider - what is really more likely here? Were you really lucky enough to pick the right door in the first place? Well perhaps...but isn't it MORE likely that you picked an empty door...and the prize is behind the other one.

Of course it is.... a 9 in 10 chance in fact. Nine time out of 10, you should swap.

Most people can intuitively see that with a 10 door problem they should swap. So think about a nine door problem...then an eight door...

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hope that is helpful

Care to haggle? How about 100 doors?

how about 50?

OK, I'll settle for 50. Monty will be out of breath by the time he's opened all those, but it will had to the interest of the entry.

I agree that it's worth writing an enry about. I find I wasn't really convinced by the entry, however. I had to experiment and then formulate my own wording of an explanation. I found that coming at it from Monty's point of view helped me to grasp the problem much better than trying to figure out the probability of my choice being a prize-winner at each step. It's not the probability of your first choice that seems important, it's the probability that Monty is choosing the only remaining empty door (rather than one of two) that determines whether a switch is worthwhile. There's always a 2/3 probability that Monty is picking the only remaining door, so, there's always a 2/3 probability that a switch will give you the prize.

A very good explanation of a a slippery problem . I liked it. How about some links to Monty Hall simulators, seeing as you mention them?

It reminds me of another statistical teaser. Suppposing your chance of having an accident when driving to work is 1% (you drive in Italy ). Then, what's your chance of having an accident after 100 journeys? It *isn't* 100%. It's about 63%. You have to raise the probability of you *not* having an accident after 100 days, which is 0.99, to the power of 100, then subtract that away from 1.

Slippery things, stats.

FM
Scout

Here's another, much twistier one.

Conside a rare, life threatening disease out there in the population.

Let's say the true incidence of these disease in the population is 1 in a million.

Luckily there is a very relaible test for this disease - a test that is 99.99% accurate (That is to say that it will give the right result in 9999 times out of 10,000 tests)

You take the test and the test is positive.

So - what is the probability you have this nasty disease?
STOP NOW and answer




Did you answer 99.99% likely you have it?
Wrong.

The true answer is, amazingly, about 1% chance that you have it. ie it is alomost certainly a false result.

Why is this, with such an accurate test?

It's actually related to the Monty Hall problem. The point is the incidence of the disease - the chance you had in the first place is 1 in million, so it's actually very rare.

What difference does that make?

Well, let's say a million people take the test
one person is positive and will test positive (ignore the small chance or error)
but about 99 of the others will test also test positive (false positive).

So false-positives are almost 100 times more common than real positives.

Moral of this
1) Statistics are twisty, counter-intuitive tests
2) Always ensure you are tested twice before believing any result. The chance of two false positives are small.

That's an interesting paradox, Botogol, but this isn't really the place for it. This conversation is about this entry, the Monty Hall Problem.

I've added a link to the very simple 'Cheap Monty Hall' simulator.

I've taken out the link to the Monty Simulator, because it wasn't what it seemed. I've also removed all reference to simulators from the entry, because all the ones I tried had problems. I've replaced it with a suggestion that you get a friend to cooperate and experiment.