Counterintuitive Maths: Part One

Cross posted at my new blog, Divisible By Pi…

We’ve all had it. You’ve just made a new acquaintance, or a number of new acquaintii (learn it, love it, use it — that plural is here to stay!), and inevitably they’ll want to know what it is you actually do with your time in real life. Being a student gets you past the first, most general conversational hurdle, but you know that eventually you’re going to have to be more specific. If you study medicine, law, or engineering you’re sitting pretty — most people have a vague to decent understanding of what’s involved in those degrees, and in a worst case scenario can at least feign interest until they finish their drink and need a refill. Biology is pretty safe unless you happen to have been cornered by the only creationist at the party. Languages give you instant credibility.

Pure maths gets mixed reactions.

In general it goes a little something like, “Oh, maths. I don’t know how you could stand that — I hated maths at school, haven’t studied it since.” And then either the conversation or the person moves on. At a basic level the issue stems from the content and style of high school mathematics; courses that seem specifically designed to suck all the fun and creativity out of maths. As someone once said (though I regret forgetting who or where), if they taught English in school the way they teach maths, kids would be assessed on filling out DMV forms.

But I digress — that particular rant is for another time.

What I’m getting at here is that maths is more than what most people realise it is. It’s creative, it’s beautiful, and it can be extremely counterintuitive — which, as the name of the title implies, is what we’re here to talk about. So from one awkward segue way to another, what better place to start talking about counterintuitive maths than with the Monty Hall Problem? For those who don’t know, the Monty Hall Problem (named after the host of the American game show Let’s Make A Deal) is traditionally formulated as follows:

You are on a game show, presented with a choice of three doors and a host who knows what lies behind each door. Behind one door is a car, and behind the other two are goats — presumably, unless you’re a particularly experimental New Zealander, you want to win the car. You choose a door, and the host then opens one of the other two doors to reveal a goat. Then, he gives you the option: Stay with your original door, or swap. What should you do?

Most people will instinctively tell you that it makes no difference — there are two doors, so either is just as likely as the other. Most people will tell you this because most people suck at reasoning with uncertainty.

The correct answer is that you should swap — the swap strategy, it turns out, will win \frac{2}{3} of the time. There are a few ways to think about this, but the easiest in my opinion is to consider the information you have first about the door you originally picked, and then about the door you’ve been offered.

First consider your original choice of door. It’s uncontroversial to point out that at the beginning of the problem you had a \frac{1}{3} probability of choosing the winning door. It should also be reasonably uncontroversial to say that, of the two remaining doors, you already know that one of them has to be a goat. What this means is that when the host reveals a goat behind one of the unchosen doors, he’s not doing anything except for telling you something you already knew — his reveal gives you no additional information about the door you originally picked, and thus logically the probability that you picked the winning door must remain the same: \frac{1}{3}.

Now, what about the other door — the one you have the option to swap to. Initially, of course, you know that it alone has a \frac{1}{3} chance of being the winning door. It stands to reason, then, that together the two initially unchosen doors have a combined probability of \frac{2}{3}. So, what happens when a goat gets revealed?

As before, this is not a surprise — after all you already knew that one of the doors contained a goat. So, from a naive point of view you still haven’t learned anything novel — the two doors, taken together still have a probability of \frac{2}{3}. But in reality you do now have some extra relevant information — because the probability that one of those doors contains the winning prize has collapsed to zero; in effect, passing all of its probability on to the remaining closed door. Thus we can conclude that the ‘swap’ door has a \frac{2}{3} chance of containing the winning prize.

This result often takes people by surprise; indeed, there are people who will refuse to accept the result as true even after stepping through the reasoning. There are two doors remaining, the argument goes, and so the chance that either one contains the car is even, a 50 – 50 proposition. Why people insist on this naive and incorrect line of reasoning after having been shown the correct answer is perplexing — one can only assume that it has to do with a gut reaction against the counterintuitive correct answer.

Realistically, of course, we probably shouldn’t be too surprised about this. People are generally pretty awful at probabilistic reasoning, for the simple reason that it generally goes against the ‘common sense’ answer we think should be correct. In hypothetical situations this generally amounts to no more than heated words and hurt feelings; in real life it can have far more serious consequences, even so much as sending innocent people to jail. The Monty Hall Problem, while contrived, is fascinating for its ability to shine a light on this major flaw in human cognition.