March 2019 – MathemaGical Thinking

One In 52 Continued

Before we explain the significance of this question you’ll need some background. I am giving one of my training courses to a group of investment bankers, hedge fund managers, risk managers, regulators, anyone with an interest in the mathematical side of finance. The audience will be mostly those with an economics or finance degree, some scientists, with the occasional lawyer. I know this sort of audience well. I know that they’ll have a pretty decent grasp of some narrow areas of mathematics and statistics, but they’ll probably wildly overestimate their abilities. And despite their six-figure salaries they won’t know when and where to apply what mathematics they do know. With this trick I’m hoping to hammer home asap that there’s more to the application of mathematics than what you find in the text books. And I’m warming my audience up. Lectures with audience interaction are more memorable than those without. And those with the invisible deck of cards are even better.

I started this segment of my course by asking the audience to imagine that they are at a magic show. I then ask someone to name their favourite card. Then by getting them to say how many cards there are in a normal deck I make the final question look like it is about probability theory, “What is the probability that…?” And people working in finance, like this audience, use probability theory and assumptions about the stock market’s random behavior as their very bread and butter. One of the most famous, non-technical, books on the mathematics of the stock market by Burton Malkiel is even called “A Random Walk Down Wall Street.”

But it’s too easy to fall back on mathematics if that’s your field of expertise. And sometimes that mathematics might not only be irrelevant, but also dangerous.

As we’ll see, context is all when it comes to mathematics, and magic.

We’d like you, fragrant reader, to take part in my exercise. Imagine you are in the audience, imagining you are at a magic show. What do you think is the probability that the card that I have chosen is the Ace of Spades? (Yes, yes, we know I didn’t really do the trick, I pretended to do the trick. So I could just as easily pretend to cheat. But we want you to imagine you are in the audience of a real magic show, and the real magician has a real deck of cards. One day I will learn how to do this trick properly myself.)

The question to you is what is the probability that the card taken from the deck is the Ace of Spades?

Think about this question while we talk a bit about risk management. Feel free to interrupt as soon as you have an answer. Oh, you already have an answer? What is that you said, one in 52? On the grounds that there are 52 cards in an ordinary pack. It certainly is one answer. But aren’t you missing something, possibly crucial, in the question? Ponder a bit more. Clue: Context.

One aspect of risk management is that of “scenario analysis.” Risk managers in banks have to consider possible future scenarios and the effects they will have on their bank’s portfolio. They like to assign probabilities to each event and then estimate the distribution of future profit and loss. Of course, this is only as useful as the number of scenarios you can think of. And you need to know those probabilities.

You have another answer already? You’d forgotten that it was a magician pulling out the card. Well, yes, we can see that might make a difference. So your answer is now that it will be almost 100% that the card will be the Ace of Spades, a magician is hardly going to get this trick wrong. That’s quite a different answer from the one in 52. Are you right? Well, think just a while longer.

Sometimes the impact of a scenario is quite easy to estimate. For example, if interest rates rise by 1% then the bank’s portfolio might fall in value by so many hundreds of millions. But estimating the probability of that interest rate rise in the first case might be quite tricky. And more complex scenarios might not even be considered. What about the effects of combining rising interest rates, rising mortgage defaults and falling house prices in America? That’s less a matter of probabilities than, with hindsight, an inevitability. And by assuming that the laws of probability trump causality leads to overconfidence that all is well with the world.

Most mathematically inclined finance people when asked the magician question, usually give the one in 52 answer – because they ignore the context, it’s a magic show. It often requires quite an awful lot of major hinting before the “quants,” the banks’ tame mathematicians, even begin to think beyond pure probability, and bring in context. Rather frighteningly, some people trained in the higher mathematics of risk management still don’t see the second answer, the 100%, even after being told. It’s as if the context is irrelevant. Or they willfully ignore the context to keep it to a nice simple question in probability theory. Heaven forbid that they should consider messy reality.

I have asked this question at many risk-management events, so I have some idea of the statistics of the answers versus the make-up of the audience. I once asked the question at an actuarial conference. Out of the audience of one hundred there were two who absolutely and categorically refused to entertain the idea of anything other than the grade-school one-in-52 answer. No amount of discussion of context and the reality of magic shows persuaded them to even entertain the possibility of another answer. One member of that audience shouted out “Those two work for a regulator!” I thought this was a joke. But it wasn’t. Seriously, the only members of the audience stuck on the mathematics, unable to see the context, were the only two from a financial regulator. Surely regulators more than anyone must consider reality rather than theory? Apparently not. These two regulators were asked to justify their answers. Their explanation involving concepts from higher probabilistic mathematics was met with hoots of amusement from the rest of the audience.

Actually there is no single, correct answer. This is really an exercise in creative thinking – and non-mathematicians are usually better at spotting this. And creative thinking is something that risk managers and regulators need to get good at. (And less of the creative accounting.)

For example, one possible answer to our card-trick question is zero. There is no chance that the card is the Ace of Spades. I usually reveal that the card I pulled from the deck is… “The Three of Clubs! D’oh!” Has the trick gone wrong?

This trick is too simple for any professional magician. Maybe the trick is a small part of a larger effect, getting this part “wrong” is designed to make a later feat more impressive … the Ace of Spades is later found inside someone’s pocket. Or, our favorite, tattooed on the magician’s arm. Very, very rarely does anyone ever think of these possibilities. (And if you did, then you should be in the Magic Circle.)

The answer one in 52 is almost the answer least likely to be correct. Magicians rarely rely on probability.

Risk management requires an open mind – but a traditional education in finance often works to close it.

So, what was your final answer?

Did you say one in 52, and stick with that answer? You are going to be one tough critic then.

Did you say one in 52 and then change your mind? Good, we can work with you.

Did you say 100%? Excellent.

Did you say zero? We don’t believe you! (You’re not David Blaine are you? We know he’s a fan but…)

Did you say 37.26%? Interesting.

Running with the idea that the magician deliberately gets the card wrong in an end-of-second-act cliffhanger there is the tiniest of probabilities that he fails…i.e. he unintentionally picks out the Ace of Spades. The correct card. And what’s the probability of that? One in 52! We’re back where we started. Does your brain hurt yet? I have never known anyone to take the analysis and the context as far as this. If you did then we definitely want to hear from you.

One In 52

PAUL

Pick a card, as they say, any card.

INT. A MEDIUM-SIZED LECTURE THEATRE – DAY

PAUL, the lecturer, is standing at the front of an almost-full lecture room. The audience is almost entirely male. But PAUL is facing a female, seated, member of the audience.

PAUL

Do you want to change your mind?

She shakes her head.

PAUL(CONT’D)

Are you sure?

She nods.

V.O.

People never do change their minds. It doesn’t make a blind bit of difference to the trick. But still it would be nice if occasionally…

PAUL

Thanks, you can sit down. You’ve just told us that the Ace of Spades is your favourite card. It was your free choice, correct?

Without waiting for an answer PAUL walks along the front.

PAUL (CONT’D)

I need another volunteer. (Pause) You!

He points at a man on the front row.

V.O.

At school the troublemakers sit on the back row. With this crowd they sit at the front. It doesn’t matter. No one can mess up this trick.

PAUL mimes throwing something to the man.

PAUL

Catch!

The man mimes catching it. He looks sheepish. There are a few muted laughs among the audience.

PAUL (CONT’D)

That’s an ordinary deck of cards, right? Look through the deck. All different?

The man half heartedly mimes looking through the imaginary deck. PAUL looks into the audience.

PAUL (CONT’D)

How many cards in an ordinary deck of cards?

AN AUDIENCE MEMBER

Fifty two!

PAUL

Right. That was an easy question. Don’t worry, I won’t ask difficult ones. Trick questions maybe…(he looks back at the man with the invisible deck) Now shuffle the deck.

Both PAUL and the man mime shuffling a deck of cards. The man is warming up.

PAUL (CONT’D)

And fan it. (Pause) No! No! Turn it the other way around, I don’t want to see the cards!

The audience laughs. They’re warming up…at last, PAUL thinks. PAUL reaches out with his right hand into the imaginary deck and mimes moving his fingers back and forth among the cards. His fingers stop, and slowly he mimes pulling out one of the cards. He brings it close to his chest, hiding it from view with his left hand. He looks at the imaginary card, and then at the man.

PAUL (CONT’D)

You can put the cards down now, thanks.

He looks at the audience and nods slowly.

PAUL (CONT’D)

Now a slightly harder question. What is the probability that the card I have in my hand is our friend’s favourite, the Ace of Spades?

To be continued…

Swingers

This post looks at Mathematical Thinking, which we define as “The belief, especially characteristic of scientists, that events can be understood, described, and predicted using mathematical equations; thinking founded on this belief.” Sometimes it’s a good thing, sometimes not so good.

Imagine you’re at the bar, and order a gin martini. The bar man serves it in the usual conical martini glass. Tell us, when the drink is halfway down the side of the glass how much is left?

Half? No, the glass is pointy at the bottom so there’s clearly less in the bottom half than in the top half. A quarter? Final answer. One quarter!

Wrong.

You may recall from school the idea of similar triangles. They are triangles with the same shape, but a different size. This is what we also see in the cone, our martini glass. The bottom half of the martini glass is “similar” to the full glass.

The problem of calculating volumes of geometric objects goes back to the ancient Greeks. Draw a square on a piece of paper and then draw another with sides twice the length. How much has the area increased by? It’s four times bigger. You can see that four of the smaller squares will fit inside the larger. Now take eight sugar cubes or dice. Put four into a square and put another four on top. You’ve now got a larger cube, each side of which is twice the length of the smaller. So for a cube halve the length, you’ve got a volume that’s a mere eighth of the original!

The shape doesn’t matter here. All that matters is that the half-full glass is “similar” in the mathematical sense to the full glass. The formula is just 1 divided by 2 raised to the power d, where d is the number of dimensions. Since we live in d = 3 dimensions – we’re leaving out spooky extra dimensions – the answer is 1 divided by 2 cubed, which is 1/8, or 12.5%.

Or in mathematical terms: it’s time to order another.

Mathematicians are full of tricks like this where pure thought can lead to answers that are not otherwise obvious or where the brain gets tricked. Say that, reaching for a bottle on an upper shelf behind the bar, the barman’s arm brushes against a light hanging from a cable, making it swing gently. What is the time taken to swing left to right then back again (i.e. the period)?

Galileo first became interested in this problem when he noticed the sway of a chandelier in the Pisa cathedral. There are a few ways of solving it mathematically. Most require some knowledge of physical laws, but there is a simple method known as dimensional analysis that gets you most of the way there. As the name suggests, it is again based on the concept of dimension.

We first ask what quantities could affect the period of the swing. A couple are obvious: the mass of the weight, and length of the string. Anything else? The colour of the string or the time of day shouldn’t come into it, but one can imagine the answer is different on the moon from here in the bar so gravity must play a role. The acceleration due to gravity, usually represented by g, is 9.8 meters per second squared. On the moon it is only about 16 percent of that number, because the moon is less massive.

Now let’s postulate that the period is given by multiplying or dividing those numbers together, where again you are allowed to raise a number to a power. We will include fractional powers, so for example a number raised to the power ½ is the square root. The trick is that, for the formula to work, the units of the expression of the period, which are seconds, have to be the same as the units of the expression that you just made up. The two sides of the equation must be made of the same stuff.

The two sides of an equation must be made of the same stuff.

Write it out and you find that the only way to make it work is if – ta da! – the period of the swing is proportional to the square root of length divided by the gravitational constant g. So the mass doesn’t – and can’t – come into it.

There is no magic involved here, no spooky automatic writing or communication with the other side, but in a sense the result is magical, because it shows us something very counterintuitive, namely that the time of the swing doesn’t depend on the mass of the pendulum. The weight only comes in through the acceleration due to gravity. On the moon gravity is less and the swing will take longer. In fact, you can use this relationship to test tiny local perturbations in the gravitational field.

It turns out that if the length is exactly one meter, then the period T is almost exactly 2 seconds, and a single swing counts out a second. Which is why traditional grandfather clocks, first produced in the late seventeenth century, have a pendulum of this length. It also inspired the definition of the metre a century later.

This technique of dimensional analysis has been used to great effect in other circumstances. New Mexico was the site of the first explosion of an atomic bomb, at 5:29am on July 16, 1945. Codenamed “Trinity,” much of the project – including the amount of energy released by the explosion – was obviously kept secret. However one observer at the explosion was the mathematician GI Taylor. Taylor is famous for his work in fluid and solid mechanics, in particular the difficult problem of turbulent flow. He estimated the energy in the atomic explosion by applying precisely the above dimensional trick but with quantities energy, air density, radius of the explosion, and time. He estimated the energy released to be 17 kilotons of TNT. President Truman later revealed it to have been 22 kilotons. Taylor’s approximation was remarkably accurate and shows the power of quite simple mathematics, or perhaps more precisely the power of Mathematical Thinking. As we’ll see, similar thinking was behind the bomb itself – this is a dangerous kind of power.

Few people are as clever as GI Taylor. And unfortunately there are many people as gullible as Doyle. But there is also an intersection between magical thinking and mathematical thinking where the power and beauty of mathematics gets used as a decoy to confound while giving the illusion of clarity. We first discovered this happening in finance and economics. But once you’ve been alerted to the possibilities of such abuse you start seeing examples everywhere.

This is what we call Mathemagical Thinking. And in the wrong hands it can be as dangerous as nuclear weapons.