Perhaps the most useful thing you will read today…

The risks of risk management“, Paul Wilmott, BBC, 5 December 2008 — Hat tip on this to Brad Delong.  Excerpt (reformatted):

You are in the audience at a small, intimate theatre, watching a magic show. The magician hands a pack of cards to a random member of the audience, asks him to check that it is an ordinary pack, and to give it a shuffle. The magician turns to another member of the audience and asks her to name a card at random. “Ace of Hearts,” she says.

The magician covers his eyes, reaches out to the pack of cards, and after some fumbling around he pulls out a card. The question to you is what is the probability of the card being the Ace of Hearts?

Oh, you already have an answer? What is that, one in 52, you say? On the grounds that there are 52 cards in an ordinary pack.  {FM note:  sometimes there are 52 cards in a pack; sometimes there are one or two jokers.  Assumptions are IMO the most common source of errors.}

It certainly is one answer. But aren’t you missing something, possibly crucial, in the question?  Ponder a bit more. … Assign probabilities to each event and you can estimate the distribution of future profit and loss. Not unlike our exercise with the cards. Of course, this is only as useful as the number of scenarios you can think of.

You have another answer for me already?

  • You had forgotten that it was a magician pulling out the card.
  • Well, yes, I 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 Hearts – the magician is hardly going to get this trick wrong.

Are you right? … Are those the only two possible answers?

… When I ask finance people this question, I usually get either the one in 52 answer or the 100% answer.

  • Some will completely ignore the word ‘magician,’ hence the first answer.
  • Some will say “I’m supposed to give the maths answer, aren’t I? But because he’s a magician he will certainly pick the Ace of Hearts.”
  • Rather frighteningly, some people trained in the higher mathematics of risk management still don’t see the second answer even after being told.

There is no correct answer to our magician problem. The exercise is to think of as many possibilities as you can.  For example, when I first heard this question an obvious answer to me was zero. There is no chance that the card is the Ace of Hearts. 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 Hearts is later found inside someone’s pocket.
  • Or maybe on the card are the winning lottery numbers – which are drawn randomly 15 minutes later on live TV.
  • Or maybe the magician was Tommy Cooper.

When I ask non mathematicians, this is the sort of answer I get. Once you start thinking outside the box of mathematical theories the possibilities are endless.

Magicians And Mathematicians“, Paul Wilmott, posted at Wilmott ( the leading resource for the Quantitative Finance community),12 December 2008 — Excerpt:

A member of didn’t believe me when I said how many people get stuck on the one in 52 answer, and can’t see the 100% answer, never mind the more interesting answers. He wrote “I can’t believe anyone (who has a masters/phd anyway) would actually say 1/52, and not consider that this is not…a random pick?” So he asked some of his colleagues the question, and his experience was the same as mine.

He wrote “Ok I tried this question in the office (a maths postgraduate dept), the first guy took a fair bit of convincing that it wasn’t 1/52 !, then the next person (a hardcore pure mathematician) declared it an un-interesting problem, once he realised that there was essentially a human element to the problem! Maybe you have a point!” Does that not send shivers down your spine, it does mine.


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5 thoughts on “Perhaps the most useful thing you will read today…”

  1. But it is because this is not a pure maths problem that it is interesting. It just underlines why so many mathematicians are incapable of getting probability.

  2. Please consider extending this issue to all ‘economic laws’ — which are merely explanations of how people actually behave. And people, unlike gravity (so far as we know in our little area of the space-time continuum), are willing and able to change their behavior.

    Usually economic laws are such that nobody can easily make money by relying on the law. ‘Supply and Demand determine Price’ is true, but how does that guide one’s behavior?

    Whenever there seems to be an economic ‘law’ that people can use to make money, there will be at least two big effects.
    1) More and more people will follow that law to make money (i.e. buy tulips; buy internet stocks; buy US houses), until
    2) The ‘law’ is no longer true.

    That is why economics is more art than repeatable science.
    This despite my MS in Engineering-Economic-Systems, where I applied engineering type analysis to engineering systems problems, in models, and then to similarly modeled econ systems.

  3. I have had experiences like this with my Uncle who is a statistic professor at UC San Diego. I am an engineer working in the oil and gas industry, and he asked me why we don’t simply do an analysis on the wells, rocks, etc. to determine where to drill next. I tried over and over to explain to him two principals – that there was nothing statistically useful to gather data about, since the underground is heterogenious and consists of many different types of rocks, fluids, etc. and that the cost of gathering some of the data was so great that it was cheaper to just ‘go out and try’ than it would be to gather to data to do it right the first time. He just couldn’t understand, and eventually I just gave up trying to explain it to him. This in spite of multiple real world examples of the sorts of problems (and solutions) that I face on a daily basis.

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