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

This entry is almost identical to Atlantic Cable's one. What are youse guys up to? Spreading doubt and confusion?

Hi Recumbentman! I wasn't subscribed to this, because it isn't finished yet, so I've only just noticed your comment. I wrote this piece last December, but wasn't fully happy with it. It is factually correct but not explanatory enough, so it needs more work. I'll get around to it eventually. Atlantic Cable wrote his one later. I doubt he copied mine, but he linked to it, which is strange. He more or less says, that his is an entry about a topic which is much better explained in mine.

I have serious reservations about Atlantic Cable's version of this. I must tidy up and submit mine before he puts his into Peer Review.

"There are 'Monty Hall simulators' available on the web where you can try your luck by choosing and sticking or choosing and changing. A few hundred losses will convince you that you are better off to change."

If there are, and they do show the probability changing despie nothing happening to affect the probability, I would suspect their veracity. I cannot accept that a mathematically versed person like yourself believes that probabilities can change, while simultaneously not changing.

What I take exception to is "Since Monty opening a door can't affect whether there is a prize behind your door or not, the chance must still be 1 in 3 after he has opened the door."

The chance is not still 1 in 3. Your having chosen it didn't fix its chances forever. You know that. What are you driving at?

R, the probabilities are as I have stated them. Which means that if you don't understand it, I mustn't have explained it properly. This will require further thought.

Hmmm...

This is a nightmare. Wake me up someone.

Do I detect a smidgeon of disbelief?

You're not alone. As I said in the entry, mathematicians have almost come to blows over this. But it's correct.

Atlantic Cable says that the paradox was devised by the smartest person in the world. I know she brought it to the public's attention, but I'm not sure that she originated it, so I didn't mention her in my entry.

"Do I detect a smidgeon of disbelief?"

And then some.

"You're not alone. As I said in the entry, mathematicians have almost come to blows over this. But it's correct."

Now tell me. How can it be correct? Three doors A B C. I choose A, Monty reveals B, C (and only C) gains in probability. Am I missing something here? Had I chosen C would A have been the only one to gain in probability? How can my choice-history make that happen? That's the Gambler's Fallacy in its full frontal nudity.

Here's my innocent interpretation: I choose A, Monty reveals B, both A and C gain equally in probability. Please demonstrate my error.

"Atlantic Cable says that the paradox was devised by the smartest person in the world. I know she brought it to the public's attention, but I'm not sure that she originated it, so I didn't mention her in my entry."

Oh then if she's smarter than you and me it's all right and we will naturally trust everything she says. Sorry for doubting.

Have you tried one of the simulators? Would you trust it?

Sorry about mentioning the smartest woman on the planet. That's the 'Appeal to Authority' fallacy in action.

I woke up at about 4 o'clock this morning and spent about an hour thinking up a way of explaining it, but I haven't time to put it in at the moment. I'll try and change the entry tonight. I'll keep changing it until you're convinced. This is a good test of my explaining powers.

By the way, I've put the entry into Peer Review, but all the Peers seem to be in recess, because nobody except you has commented.

You don't teach Matheamatics in Trinity, do you? Last time I got into a probability argument with someone, he turned out to be a Trinity Maths lecturer. It was very embarassing.


For him.

Hi Recumbentman: try thinking of the problem this way:

You pick a door, and then Monty Hall asks if you want to stick with that door or choose both of the other doors. If you change, and the big prize is behind either of the other two doors, you win.

Now it's clear, isn't it, that you should switch, since you are getting two shots at the prize instead of one?

The stated problem is mathematically the same, since you know before Monty opens a door that one of them must be empty, and that Monty will open that one and eliminate it, whichever one it is.

You can simulate it yourself with three coins: turn one up heads and the other two harps, and take turns to play Monty and the contestant. Switching wins, it only takes a few minutes to prove it to yourself.

Hed

Hi, all. It seems as if most of the posting is being captured here, so:

The Monty Hall problem belongs among the family of non-problems that occur when chance, logic, probability and term definitions are blurred.

The original problem outlined does seem to give a 2 to 1 likelihood of winning by switching doors, but the original selection is not a one-in-three chance of being correct. It is the logic that is faulty.

The only thing originally selected is 'one of two doors that will not be opened', and not 'which door hides the prize'. This gives it the illusion of being a one-in-three chance of being right, when the first selection has no effect on the 50-50 outcome of the final selection. As the door hiding the prize will never be opened after the first selection, there will always be a 50-50 chance that the first selection was correct.

The problem is not one of probability math, but of definition of outcomes and logic.

Fordstowel: try it yourself with coins. Gnomon is perfectly correct, and you are wrong. If we were in the pub, I'd offer to do a demo with real money, where I put up a €1 prize and you stick, and then you put up a €1 prize and I switch.

Most people get the idea before they've lost €10.

Hed

Yay FordsTowel, you tell em!

Here's another illustration:

I pick door A and FordsTowel picks door C.
Monty opens B.
A mathematician tells us (while keeping a straight face) that we would both improve our chances by swapping doors.

I teach Renaissance Music and I don't feel embarrassed yet.

Sorry, I did try it. A simple set of Rand() and Randbetween(x1,x2) formulas in MS Excel did hundreds of choice comparisons for me.

Since the first choice is always going to be one of the two remaining choices, you will always have a 50-50 chance of winning no matter which of the two remaining doors you pick. It's as if the first choice and removal never happened. The first choices is simply irrelevant.

Hi RM, I don't know how your posting snuck in between, but thanks for a new logic slant on the mind-game.

All right, let's do it the hard way. I've written out 10 rounds, and I've put the results in my journal to keep the game honest [you guys are not to look yet, OK?].

Fordstowel, you will guess and always stick.

Recumbentman, you will guess independently, and always switch.

Recumbentman will beat Fordstowel by about 2:1

(Obviously ten guesses is not statistically enough to prove the point, but I'll go out on a limb here).

Round 1, gentlemen, pick a door: A, B or C?

Hed

Okay, I choose A. I will always choose A so you needn't ask me again.

Gnomon, here's a suggestion for how to persuade us.

You are more likely to pick a wrong door in the first choice, and therefore are more likely to be shown the only remaining wrong door than one of two wrong doors. On the occasions when you are shown the remaining wrong door, it is correct to switch. QED.

That looks so good to me that I am puzzled for the moment. I still agree with FordsTowel though that the first choice is irrelevant, and that the second choice is made absolutely from scratch.

Recumbentman, I've posted how you did in my journal, and you may look now

I'll wait for Fordstowel to post his ten guesses here before telling him how you did.

Hed

Hi Folks!

I've rewritten the entry and I hope it is more convincing now. If not, I'll have to resort to a table of each of the possible equally likely outcomes. While that will be conclusive, it will also be boring and not the sort of thing I want to put in an entry.