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.