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

But someone else didn't like the idea of one million doors; they found it too different from the three door problem to be appreciable the same problem. So I changed it to 50 doors.

Could I humbly (well all right -- blatantly) recommend my 'explanation', viz. that Monty opening a door (in a three-door game) is not always acting freely; in those cases where the guest has guest wrong ( couldn't resist that) he is compelled to open the *only remaining losing door*, pointing out the winning door by default. It only remains to notice that these (forced) occasions outnumber the alternative by two to one.

Why? Well, it worked for me.

It occurs to me that there is one problem with the solution you suggest.

If I remember "Let's Make a Deal" correctly, one of the things Monty Hall did was offer a cash incentive to get the contestant to either change his/her mind. In other words, the solution ignores the human presence of human intervention in the form of Monty Hall.

In fact, I seem to recall that Monty Hall himself was interviewed about the problem, but I can't recall his exact response. Perhaps you might make a reference to this?

The Monty Hall Problem is a probability paradox which is inspired by the behaviour of Monty Hall. It is not what Monty actually did in the television program. I mentioned that in the article, I think.

Gnomon, you're supposed to be on hols! And there's a heatwave on for God's sake!

I'm not actually gone until 4pm. I've said I'm gone so that I can ignore people and pretend I am away. But after 4, I really will be away.

Ha! Re-Man, this was the perfect reply that should clear it all up for the naysayers:

"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 enjoyed reading the bait of banter hooked in this debate! But of course, if there are 3 options... and one of them is removed... then neither of the two remaining options can have any chance other than 1/2!

Out of 2 possible, doors, the number 3 should NEVER appear in determining your Chances.

A major point of disctinction is the difference between Chance and Probability. Chance is calculated each Event, based on simple division of your numerator (you can pick 1 door) and demoninator (there are 2 doors left) as previously mentioned here. Chance = 1/2, or 50%.

Chance is science.

Probability is the application of Chance over many, UN-connected events.

The original 3 doors have nothing to do with the new selection of 2 doors, when determining your True Chances. In this case your Probability can be seen as 1/3, but your Chance is 1/2.

The lesson to learn is, try to only think of Chance when betting. Probability WILL mislead you in the short term, and this example is clearly based on short term events. If you could afford to bet 100,000 times, then probability WOULD matter - and the casino would only be beating you by about 5,000 spins

Hello Shamantiks. Time to revisit this wonderful puzzler.

Your first "naysayer answer" was what I thought originally, leading me to doubt what I now accept/understand/believe.

(Funny, it seems strange to put the word "believe" in there; it's a compelling mathematical fact that is perfectly indifferent to whether I or anyone else believes it or not.)

And yes, the mathematician can tell two people to swap guesses, and with justification. They are ditching a 1/3 choice in favour of a 1/2 choice. The fact that only one can win is relevant to a larger equation, not the equation of each individual.

I'm not sure of your longer message on the difference between Chance and Probability. What's the upshot? Change or hold?

I would like Gnomon's opinion on a similar case. You choose six lotto numbers, then somebody tells you that no numbers with a 2 in them will come up (they made the balls, and left out all the 2s). Should you now choose new numbers, even though your original selection turned out not to include any 2s?

By the Monty Hall rule, you should. But you are just as likely to pick the same numbers again; you were already in the safe zone. What is to be gained by changing them?

(I'm assuming that there is no principle for choosing numbers; avoiding the popular numbers in order to decrease your chances of sharing the prize is not to be considered in this question.)

This I think really points up the reason people find the Monty Hall conclusion surprising.

After three glasses of wine, I'm not up to calculating those probabilities, R. I'll have to come back to this tomorrow.

I can see an important difference between this scenario and the one where 999,998 doors are opened leaving yours and *one* other. I'd be interested to hear your conclusion, whenever you feel sober enough to arbitrate. I forgot to ask my actuary brother yesterday. I must also put the two envelopes one to him.