Today I gave the class the Monty Hall problem. I like it because most people choose the wrong solution and it is extremely hard to explain the right solution without mathematics (ie the toolkit of science). The verbal explanations are somehow so unsatisfactory. Try this nonsense for example, which is technically correct Hollywood gibberish [mind you, Kevin Spacey does define professor cool].

The reaction of one of my PhD students when I told her what I was going to do:

*Oh my God, you’re not going to Monty Hall them! *

*Do you have any idea the emotion that will generate? Paul Erdos didn’t get this problem and was vitriolically angry about it until it finally clicked. My husband and I nearly got divorced over it (he didn’t mention that Monty knows what’s behind the doors and will always choose to reveal a goat, then he refused to believe that those conditions mattered). You’re in a country where people may be armed… ARE YOU COMPLETELY INSANE?????*

which, when I explained I escaped alive, was followed by:

*.…but were they genuinely not worked up about it? It made me blindingly angry, and when I brought this up once at a party as an amusing story, the point being how funny it was that this maths problem produced such passion, two people at the party almost came to blows about it. These were neighbours, not maths geeks. Also really annoyed husband’s brother, husband’s best friend….. so what the hell is wrong with your Penn State students? Are they dead?*

Well no, but I could not decide what the students made of it. Is it just too weird to wrap your head around a pointless problem that is (by the professor’s own admission) very hard to wrap your head around? Next class I’ll discuss

**why**it is so hard for people to wrap their head around it. Shame no one understands that either. At least there are some hypotheses.
model studentFurther to Andrew’s casual use of my heartfelt intervention above, I e-mailed said husband’s best friend with the link to this blog as I thought it might entertain him, and, proving my point, he phoned this evening, just to complain that I had made him even THINK about the Monty Hall problem again. He definitely feels Penn students are weirdly inert. 🙂

Husband's Best FriendI am said “husband’s best friend”, and I’m *still* thinking about it, dammit.

I know I know the correct answer, but I still don’t think I actually grok* it.

* http://en.wikipedia.org/wiki/Grok

My brain hurts, and it’s not even midday yet!

Andrew ReadSo the students ARE dead/inert with respect to Monty.

I asked in class today (two days after spending an entire class on the problem), and only ONE of the 85 or so present had discussed the problem with anyone else (and his Dad did not get it). I was incredulous. They are far from divorce over it: they actually don’t care. So…

Two hypotheses.

1. I taught it wrong.

2. This problem does not interest these students.

Next year, will try it later in the course, and maybe use the history of the problem as the hook, instead of the problem itself.

But really. I mean really. Perhaps I have I stumbled on to a diagnostic that separates geeks from the rest of humanity.

model studentAndrew,

On the upside, the other possible interpretation is that you are the first person in history to explain this problem so clearly that no one was surprised about it. You are, after all a professional ideas-peddlar. If this is the explanation, do you have a recording of the class, because you could make a genuine contribution to average global (or geek) peace of mind by putting it on You-tube!

MATTHEW S HOFFMANProfessor Andrew (I consider this a casual title, enabling respect on principle but with some friendly familiarity),

You mentioned the horrid mathematics of winning in casino games and it got me wondering about the Gambler’s fallacy in regards to the Monty Hall problem. Does that come into play with the doors at all?

Also, how often is the first door ever chosen wrong? Or is that not a component of the problem?

The ability of humans to focus on the small percentage of failure versus the large percentage of success always reminds me of good deeds versus bad deeds. The latter tends to have the most effect on people.

But now I’m just inferring with bad data and that leads to goats.

Andrew ReadThe gambler’s fallacy is the fallacy that (to use an example) if you haven’t won for a while, you’re due a win – whereas in fact the chances of winning remain the same with each game/coin toss etc. This is like people who have had many sons expecting their next offspring to be a daughter. Nah.

I do not think that fallacy is why people find the Monty Hall problem so challenging. Instead, they just can not believe that, even on a single play, you are twice as likely to win by switching doors. Most people perceive it as a 50:50 problem when it is not.

The first door will be chosen correctly (ie have the car) 1/3 of the time (and so not 2/3 of the time); that is not the problem. The problem comes when Monty reveals new information (a different door which does not have a car); that’s when human intuition shorts out.

Nothing wrong with goats. But cars are better.

MATTHEW S HOFFMANThanks for the detailed explanation. My girlfriend refuses to admit to the gambler’s fallacy when playing roulette. I prefer craps anyway.

Wonder what Newton would say about this problem.

Andrew ReadThe Gambler’s fallacy is easy to explain for roulette. If it was not true, casinos would be out of business. If your girlfriend is gambling to get rich, she should stick with games of skill. That way, there is a slim, slim chance she won’t get screwed.

Newton? If the Monty Hall problem had been discovered then, would Newton have discovered Bayes’ Theorem before Bayes? Although, I guess goats were more valuable than cars in Newton’s day, in which case not switching is the RIGHT answer.