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’s an ordinary pack, and would he please 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?
Of course, if a card is chosen at random from an ordinary pack of 52 cards, the probability of it being the Ace of Hearts is 1/52. But is that really correct? What if this is not a math problem, but instead you are indeed watching a real-life magic show in a theatre? Do you really believe that the magician doesn’t know exactly where the Ace of Hearts is? Thus the “real” question is: how likely is it that the magician’s script calls for him to draw the Ace of Hearts? That’s certainly one possibility; but there are others, for instance the magician might pull the card from the pocket of an unsuspecting audience member.
A member of wilmott.com 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.
Once you start thinking outside the box of mathematical theories the possibilities are endless. […] A lot of mathematics is no substitute for a little bit of commonsense and an open mind.
I’ll get around to arguing with Wilmott in a moment, but let me first tell you about the number 52.
Numbers, of course, are abstract concepts. They don’t have to be associated with counting cards, apples, oranges, or anything else. How, exactly, do we define them in the abstract? Here’s how this was explained to me back when I was an undergraduate math student. We start from the Zermelo-Frankel axioms of set theory, and then proceed as follows.
- The Z-F axioms guarantee that the empty set exists. We define 0 to be the cardinality of the empty set.
- Consider the set whose sole element is the empty set. We define 1 to be the cardinality of this set.
- Now consider the set whose elements are the empty set and the set whose sole element is the empty set. The cardinality of the new set is 2.
- Repeat this 50 more times, and you get to 52.
Now, we don’t actually go through this procedure every time we have to use an integer number, let alone fractions. The point is, though, that mathematics deals with idealized abstractions and that we tend to be well aware of our limitations as far as real-life problem-solving is concerned. Ask me to solve the differential equation and I will tell you, with 100% certainty, that . But is this really the equation that you should be trying to solve? That’s where the mathematician needs to hear from someone who actually understands the context. Pure mathematics, alone, cannot speak on that matter.
Wilmott is right to say that it is a problem when mathematics gets to overrule common sense. His diagnosis of the underlying causes, though. gets it exactly backwards. The problem isn’t limited to applications of mathematics, either. Here’s the actor Philip Seymour Hoffman talking about his latest movie Doubt:
“What’s so essential about this movie is our desire to be certain about something and say, This is what I believe is right, wrong, black, white. That’s it. To feel confident that you can wake up and live your day and be proud instead of living in what’s really true, which is the whole mess that the world is. The world is hard, and John is saying that being a human on this earth is a complicated, messy thing.” Hoffman paused again. “And I, personally, am uncomfortable with that messiness, just as I acknowledge its absolute necessity. “
And that’s the real “human element” at work. Uncertainty and doubt have been a part of the human condition from time immemorial, but so has our discomfort with them, our struggle against them. We want security and certainty. We long to be reassured – by religion, medicine, mathematics. We want to be told what the future will bring and we want a 100% refund in the unlikely event that the prediction fails.
There’s a sense of security in having a formula that lets you make predictions. You get to print nice glossy brochures with charts, graphs and tables, citing scientific publications in top journals. The formula, though, is only as good as the assumptions that went into setting it up. If those are true – every single one of them – then the mathematically predicted outcome is inevitable. Just like we would like it to be. We don’t want to read the fine print.
In practice, there’s usually at least one unspoken assumption that fails to hold, namely that the system in question is isolated and there are no more variables to be taken into account. Sometimes it’s reasonable to consider the system as if it were isolated. Other times, it’s not. How do we know? Maybe, really, we don’t.
We don’t like to worry about it, though. We prefer to accept the mathematical solution to the easier version of the problem. I’ve seen it in every calculus class I’ve taught. The “word problems” are grossly simplified versions of real-life situations, simplified so that the problem can be solved using first or second-year calculus. Most of them are worded so as to make it clear which formula should be used, and if that’s not obvious right away, it will be after several repetitions in class and on the homework. When I try to ask the students to consider what unspoken assumptions are being made or in what data range the solution will no longer be correct, it tends to fall on deaf ears, because that’s not a part of the problem, is it? Open-ended problems – find a good approximation for something or other, using your common sense to determine what’s “good” – won’t get me far, either, because how exactly am I going to grade it on a test?
Mathematics is not the enemy of common sense. Intellectual laziness is.
Oh, and those arrogant mathematicians from Wilmott’s story who didn’t “get it”? Sounds too much like a ratemyprofessor complaint. “The first guy” might have been told that this was “a math problem” in a way that suggested strongly that the human factor should be disregarded, as it always is in those calculus problems I’ve just mentioned. The “hardcore pure mathematician” may have been immersed in his work, as we often are, and did not appreciate the interruption. Or he may have suspected one of those “mathematician jokes” that paint the mathematician as a real-life idiot and make the layperson feel oh-so-glad that he’d never learned algebra. Sure, there are mathematicians who live in an ivory tower. There are others who don’t. Consider all scenarios and do not make unwarranted assumptions.