Heads... Heads... Heads...

Bet?

Heads I win.

... Heads.

A weaker man might be moved to re-examine his faith, for nothing else at least in the law of probability. One: probability is a factor which operates within natural forces. Two: probability is not operating as a factor. Three: we are now held within, um... sub or supernatural forces. Discuss!

I first came across the Monty Hall problem in Fermat's Last Theorem. Its goes like this:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the other doors, opens another door, say No. 3, which has a goat. He then says to you, 'Do you want to pick door No. 2?' Is it to your advantage to take the switch?

A reader asked Marilyn Vos Savant's Sunday Parade column this question in 1991, and Marilyn - a woman with an apparently super-human IQ - answered yes: it's better to switch. If you switch, you have a two thirds chance of getting the car. Most people - including me originally - think it's 50-50, and therefore it doesn't matter if you switch or not. We're wrong.

At the time, I couldn't get my head around the logic, but just presumed this would go away if and when I applied myself to the problem. It has since re-surfaced several times, and my intuition has each time failed to accept the result.

Most recently it came up round the dinner table at the Complexity conference over the summer. A guy from Edinburgh had actually managed to get a little bit of cash from some friends by playing the game with them. (This is a fairly safe money-making proposition: switching rates for this or similar problems are reported as being between 7% and 17%. People also - even though if they agree that it's 50-50 and so doesn't matter if they switch - tend to prefer their original choice.)

I said I still couldn't get it, so he explained it in depth. Everyone else entirely got it. I still didn't. I'm afraid I have to confess that I said I did. (I could kind of see the argument murkily, but couldn't 'get it' in a satisfactory way.)

We then got into a discussion on the nature of 'grasping' mathematical problems and whether the shift is sudden or not. For me, often it isn't: it's either a 'bar of soap in the bath' phenomenon, or I get to it through an onion skinning process that seems to reveal deeper layers of 'grasping'.

It came up again this weekend. Me and Sue ended up getting some cups and coins and testing it. This took a fair few runs, but we produced enough of a result to be satisfied that the probability theory wins.

I haven't been able to stop thinking about (and sometimes shouting about) this problem for three days now. Why? Well, before I get on to the rant, first lets go over some of the many ways to help the intuitively challenged like myself.

1. from Kevin McConway's Mathematical Thinking pages on the Open University site.

One way to think of the problem is in terms of three different scenarios. There are three ‘prizes’: the car, goat 1 and goat 2. The possible scenarios are as follows.

1. The contestant originally picks goat 1. The host shows goat 2. The contestant will win the car by switching.
2. The contestant originally picks goat 2. The host shows goat 1. The contestant will win the car by switching.
3. The contestant originally picks the car. The host shows either one of the goats. The contestant will lose by switching.
These three scenarios are all equally likely (because the contestant has no idea where the car is, originally, and is making the original choice at random). In two of the scenarios, the contestant wins by switching. In the third, the contestant loses by switching. So the contestant has two chances in three of winning if they switch, and only one chance in three if they don’t switch.

Looking at it this way also shows what’s wrong with the argument that there are two possibilities after the host has opened the door — car behind door 1 and car behind door 2 — and that these are equally likely, so that it doesn’t matter whether you switch or not. There are indeed two possibilities, but they aren’t equally likely.

Yup - still two possibilities. But not equally likely. Um... Here's the one that helps me think about it best:

Initially, there’s one chance in three (probability 1/3) that the contestant chose the right door, and two chances in three (probability 2/3) that they didn’t. Whatever the contestant chose, the host can open a door with a goat behind it, so the fact that the host did this does not affect those probabilities. So the contestant can stick with the original choice (door 1), and they’ll still have a probability of 1/3 of winning the car. Or they can change and say that the car is behind one of doors 2 and 3. What the host’s action has told them, that the contestant did not know before, is which of the remaining doors might have the car behind it. It can’t be behind door 3 now, because that has a goat. So the 2/3 probability of getting the car, that originally applied to doors 2 and 3 taken together, now applies just to door 2, and the contestant should switch.

Magic! Er, no: probability. It does, in a way, make perfect sense, doesn't it? There's also the example where there are 100 doors, you pick one, and the host reveals 98 goats - leaving you again with two. So there's only a 1 in a hundred chance of getting the goat, but now you have a 99 in a hundred chance of success if you swap. When I visualise this, though, I still don't get it. I'm still left with two doors, two choices, even split. Oh good God.

On a more manageable scale, you could do the same with four or five cups: if the 'host' then reveals where the other lower-value coins are - leaving you only the choice of sticking or changing - it becomes intuitively easier to accept that changing is better. Well, sort of.

Another method - which is just the 'fucking get used to it' option of last resort - is to do it over and over until you 'accept it'. I'm starting to get that - but this is just learning not to worry about it.

The original link above also has a nice little probability wheel that gives a visual representation of what's just been said.

It also goes on to show what role the knowledge of the 'host' has in the outcome of the problem: there's another wheel where it's shown what happens if the host has to choose randomly from the remaining two cups: the probability returns to 50/50. Which is to say, on the times where the 'host' chooses a goat still, it doesn't matter if the contestant switches or not.

So - I've stared at the problem, and when I look at the numbers, it makes sense in the abstract. But its given me more ontological angst than I've had in a long time. I have felt not dissimilar to poor ol' Rosencrantz and Guildenstern. I stare at those two remaining cups, and I think:

Two cups. One penny. One five pence piece. This cup: somehow it's got '1/3' under it, and somehow the other has '2/3s' under it. Until I lift one. Christ, maybe I'll find a dead cat as well...

Of course, no-one in the world has more privileged access than anyone else to whether a particle has decayed (if we put aside that some people have acccess to electron microscopes and the like, but you know what I mean...) With the cups, the host does know.

What am I saying? Are those coins under the cups in a superposition of 1/3 and 2/3 states? Only for me. And anyway that's nonsense: it's an entirely classical system, innit? Though if quantum mechanics is only a probabalistic system because we can't measure any better, then maybe its not that different...?

I think it can be argued that's the case. Maybe those probabilistic states are fundamental to the nature of the universe...? No they're not - they're fundamental to the nature of our ability to know the world, which isn't the same thing.

What if I've chosen a cup, the host has revealed a penny to me - and then someone else comes in the room and is asked to choose? Do they suddenly impose a 50/50 superposition on the coins?

Weelll - yes and no. This is where it becomes clear that, not only is the knowledge of the host necessary, but time is also. The 50/50 and the 1/3 - 2/3 probabilities aren't intrinsic to that moment - that's an illusion (more on the 'illusion' in a moment.) It will only manifest itself if repeated.

I can get that, but here I am again, staring at the cups, thinking: 'so its a function of time. So - my intuitive understanding of time is wrong. My imagining of the coins under the cups is wrong. The maths is unutterably right. My construction of the world is a thin, pale, wavering nothingness on a sea of probability. Fuck, I'm not even sitting in this chair! It's just a spewing interaction of strong and weak forces!'

If anyone's reading this, have you fallen asleep yet? I'm going to go on and on about this, like I have to several other rather bored-looking people, which has felt mildly Kafkaesque. "Aaargh! There's something fundamentally wrong with the universe! Maths has broken it! Aaaaarrgh!"

I can see I originally have a 1 in 3 chance, and that the revealing of a coin doesn't change that. But when I look at the cups, I can't see past 'two cups, two coins - my fucking choice! My fucking bastard 50-50 choice!"

But you could have chosen the one that's just been revealed to you...

BUT I DIDN'T! AND THESE ARE THE TWO THAT'S LEFT NOW! AAAAA!

I'm not alone in this. Burns and Weith [1] tell of Erdos' refusal to accept the result. Erdos...

... one of the greatest mathematicians of the twentieth century and a man who lived to solve mathematical problems... insisted the answer was incorrect, and could not be convinced even when it was explained to him in the language of mathematics. Eventually, a computer simulation convinced him that switching was correct, yet he remained frustrated by his inability to intuively understand why. This peerless authority on probability was only mollified when several days later another mathematician friend made him see his error.

Burns and Weith, sadly, don't relay how they did this, but they conclude this...

illustrate[s] an important point about the MHD (Monty Hall Dilemma), that knowledge about probability is useful but not sufficient for solving the MHD.

(See below for another paper that shows how cognitive ability does affect the outcome.)

Burns and Weith have a few other explanations for this 'cognitive illusion'. (This is a new concept to me: a cognitive parallel to optical illusions. I'm currently working on the basis that this is what I'm suffering from, but I reserve the right to have a tantrum again and make sweeping claims about the nature of knowing, rather than slight my own dysfunctioning brain. Sadly, the research is against me...)

They argue that people have a problem grasping the causal structure of the problem, which they claim is like the 'collider principle' - the opposite of 'correlation is not cause', in the sense that there are two independent variables that impact on one dependent variable. E.g. if your car doesn't start, is it the battery or the gas tank? These two become 'dependent conditional' on whether the car starts. If one is full, it must be the other.

In my cups-with-two-pennies-and-a-five-pence-piece MHD case, placement of the five pence piece and my initial selection dictate which cup the host reveals.

I'm writing these words down, but I don't get them! In fact, I think I'll just drop that avenue for now and try something else...

There are a few papers on the MHP and quantum theory, which unsurprisingly are totally beyond me. But here's another one with an interesting suggestion: powerful heuristics over-ride other cognitive approaches. In this case, the rule is 'if the number of alternatives is N, then the probability of each one is 1/N' - failing to take the 'knowledgeable host' information into account. [2]

(Just had a random thought: if the hypothetical third person comes in to the room when there's two left, but this time the game player has labelled their original choice, this contains information too. The new player knows that choice originally had a one in three chance of being right. So it's not just the host who has knowledge. Ooo, I'm nearly getting it! Oh... oh... no, there goes the bar of soap in the bath again...)

This use of heuristics, the paper notes, has been called 'the fundamental computational bias in human cognition' which, in this case, ends up with a cognitive illusion. Yikes. It goes on to say that people who do reason correctly tend to have higher working memory capacities than the rest of us suffering from 50-50 syndrome, and its this that helps over-ride the heuristic.

I don't know precisely what 'working memory capacity' means - it seems to be like a mini random access memory for a particular cognitive component, if I'm reading the paper right. But I do know that I can't get out of seeing two cups, now, with two coins. I have enormous difficulty trying to play out a time-series of different choices and seeing what that would mean. (Though at this stage, I now cling to 'I chose this cup, and it had a 1 in 3 chance, and that ain't gone away, so that last cup must have a 2 in 3 chance...' - which is now just another learned heuristic.)

Anyway, their working memory capacity test (a Gospan test of simultaneous word recall and maths solving problems) gives a mean score of 31.94 out of a possible 60 for their whole group, and 38.08 for those giving a correct response to the MHD problem. (The paper also goes on to burden people's working memory capacity whilst they're doing the MHD, and shows they do significantly worse: they do this to get around any problems of correlation between working memory capacity tests and other possible causes. Its also interesting because they're suggesting that there are two separate processes at work - the heuristic and the ability to calculate the correct result - and that one will come out ahead when the choice is made.)

That will have to do for now, I think. At some point, probability theory is going to have to play a part in my PhD, so perhaps it'll be time to return to it then.

So, to conclude: I could be continuing not to get it because I can't reason successfully - though after this amount of time that a) seems unlikely and b) is overtaken by a new heuristic anyway.

Or, I am sort of getting it, but I'm annoyed by a different aspect of the problem, which is the relationship of informaton, cognition, probability and time. In some sense, the cups do 'hold' the probability, but only in a way that includes both time, all the other options and the already stored knowledge of another information holder. Oh, bugger. Who knows?

Or, to sum up a little more: its coz I'm either a bit dumn with this problem, or dead deep and profound and metaphysical. Hmm... well, I'd prefer the latter, given the choice, but alas I may not have that choice...!

Another possibility: I get the probability theory in the abstract. I don't get it when I'm staring at the cups. These are two separate realms of mind, and I'm trying to get the buggers to merge unsuccessfully, so's it feels intuitively right. Can't.

I'm also going to hold out the hope that somehow, I might stumble upon an entirely new probability-powered energy source.

Oo, you can almost hear the little quantum 'pop' of the probability wave collapsing when the cup's lifted...

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[1] Causality and reasoning: the Monty Hall dilemma, Bruce D Burns and Mareike Wieth (2003) from Proceedings of the Twenty-fith Annual Conference of teh Cognitive Science Society , pp.198-203

[2] Working memory capacity and a notorious brain teaser: the case of the Monty Hall dilemma, Wim de Neys & Niki Verschueren in Experimental Psychology 06: Vol. 53(2) pp.123-131