I remember one time trying to explain to this guy (with another guys help) why switching in the Monte Hall problem doubled your chances to 2/3 and eventually failing to convince him.

He had heard about how some game theory problems turn out differently if they were iterated. So no matter what approach we took to explain it, he found a way to interpret the explanation in such a way, so that it implicitly assumed iteration. Apparently, the mere utterance, "...blah, blah, blah x of the time..." was enough for him to reject an explanation and cling to his preferred 1/2 assessment for switching instead of 2/3.

It took awhile to figure out that this was going on, because he articulated himself rather poorly. In retrospect, it's rather funny, but at the time, it was quite frustrating.

A carefully constructed proof would have no possibility of convincing most people. What I'm saying should convince most anyone who understands that the question is asking for the percentage of the time that your original choice is right out of the times that Monty opens a door and shows you a goat. Since those times are all times, and since you are right 1/3 of all times, then you are right 1/3 of those times. Anyone who can't understand that is likely riding the short bus to school. I believe that percentages are taught to most 10 year olds. Come on.

Meh. I am sure a mathematician could carefully construct plenty of complicated proofs that leave most non-mathematicians perplexed.

Isn't that what I said?

Yes. But that isn't what I read.

All of this is true, but it's not what you wrote in the quote that I was objecting to . You've elaborated considerably on the quote that was meant to be convincing in of itself (which is my point here). Originally no mention was made of sets of games, "in all games" , premises following from conclusions, etc. etc.

The key to this problem is the specific detail "Monty opens a door which he knows to hide a goat". Without that detail , the answer is 1/2. It's easy to see why this could be left out in a translation by someone who didn't already grok the correct problem and solution.

As someone else pointed out though, it's extremely perplexing when people do seem to understand this detail but still come up with 1/2. Apparently this was discussed in a newspaper column once, and "approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine claiming that vos Savant was wrong." I guess one can only shrug...

It's easy to see how someone can be confused by whether or not it makes a difference that Monty chooses at random or not. I've observed (when trying to win arguments in the past!) that even if you present someone with what seems to be an irrefutible argument, if they have a different argument in mind that gives a different result which they cannot refute either, then they just ignore your argument. They won't be convinced unless you can actually refute their own faulty argument (i.e. in this case, explain why it matters if Monty can see the goats, without trying to argue "it must matter because otherwise you get the wrong answer").

Here is the original question:

It was titled "Game Show Problem". To someone who was not familiar with Monty Hall, it was less than clear that a goat would be revealed every time. And to those who were familiar with Monty Hall, they were making an assumption that was not clearly stated in the question. How the question is asked is critical for a single solution.

The original quote said:

"If you did this a million times and always just ignore what the host shows you, staying with your original choice, then obviously your choice will be right 1/3 of the time, and wrong 2/3 of the time. So if you switch every time, you will be right 2/3 of the time and wrong 1/3 of the time. "

This is convincing by itself since it is understood that "always" and "every time" that you do this, Monty will have opened another door to show you a goat. But to make this explicit, we can replace the above with:

"Monty opens another door to show you a goat in all games. Your original choice will be right in 1/3 of all games. Therefore your original choice will be right in 1/3 of all games in which Monty opens another door to show you a goat."

You left out the critical part of the story. The question (mailed in by a reader) and the Vos Savant answer were published together, and in her answer Marilyn filled in the missing assumption, saying that the host knows what is behind the doors and never reveals the prize. There was nothing missing or ambiguous in the question/answer as printed, and no excuse for the arguments that ensued.

There was an excuse -- many claimed that she was not answering the question that was asked. It's an excuse. Not a good one, I would agree. I found it amazing that some of those people would sign their name to what was obviously a knee-jerk response.

Fair point. But she did answer the one that the questioner intended to ask, and she clarified it for him.

Well i think i agree with the quote
I think once u have got the problem u must state and get it to the bottom and wring the worst thing in it and destroy the main reason in it. i mean i m saying because i have faced many problems in life and the reason i survived is because i faced them and gained or u can say learned a lot from those problems.

I, uh, what?

This is what made the light bulb come on for me, my stats professor used a deck of cards.

I have had 3 or 4 friends who didn't "get it" and after I used a deck of cards they all got it.

Basically have them pick 1 card from a deck face down and keep it to the side, telling them if they picked the ace of spades they win.

After they pick their card I then pick up the remaining deck and begin riffling through it (they can see that I'm looking at all the cards). Eventually I select 1 card to keep and turn the remaining 50 cards over and spread them all out, so that they can clearly see none of those 50 cards are the ace of spades.

At this point I ask them who is more likely to have the ace of spades, me or them.

Every single one of them then understands.