Just do the math

I am currently reading “How Not to Be Wrong” by American mathematician Jordan Ellenberg.  Not that I am wrong very often, of course, but it pays to protect one’s elevated position.  In truth, I scraped by in my maths A-level, which I took simply to please my mathematician father, and have since taken the precaution of marrying an engineer (who is therefore very numerate) so that never again will I have to work out the tip on a restaurant bill in my head.  As my grandma used to say, why keep a dog and bark yourself?

And yet – here’s irony for you – I find myself working, albeit tangentially, in the financial sector.  With numbers.  And risk calculations.  And – my personal bête noir – probability.  To be honest, I like AML in part because I can take an extreme position: if it seems too good to be true it probably is and you should STEER CLEAR – sound the klaxons and back away from that dodgy deal!  My cowardice and financial timidity protect me but every day people who are smarter than I am fall for frauds and scams.  I had always assumed that it was their vanity or inattention or overweening greed that was their undoing, but my new pal Jordan has shed some light for me on what might be happening, with his parable of the Baltimore stockbroker.  Forgive me if I am late to the party with this one, but it’s been a revelation to my mathematically-challenged brain.

Imagine you get an unsolicited email from a stockbroker in Baltimore.  (I daresay he could be anywhere, but maybe his Baltimore-ishness is important.)  You scan the email out of idle curiosity and it predicts that a certain share will rise in value this week – and you happen to hear that it does indeed gain quite a bit.  The next week another email arrives from our Baltimore friend, this time predicting that another share will tank – and again it does.  This goes on for ten weeks, and you have now amassed ten accurate share tips.  In week eleven, the stockbroker offers you the chance to invest via his service – for a generous fee, of course, and with all the usual warnings about shares going down as well as up, but reminding you about his unbroken run of ten top tips.  You do the maths to be sure: each week he could have been right or wrong with his tip, and the chances of being randomly right for ten weeks in a row are ½ x ½ x ½ x ½ x ½ x ½ x ½ x ½ x ½ x ½ = 1/1,024.  That’s just so unlikely – he must have a system – and so you’re hooked.

But, counsels Jordan, it can all be explained.   In week one, the stockbroker – who knows his maths, the fiend – sends out 10,240 emails: half predict a share price increase and the other half a share price fall.  The 5,120 people whose emails turn out to be wrong never hear from Baltimore again.  The 5,120 with the correct predictions are contacted in week two – again, half are told of a price rise and half of a price fall.  The stockbroker checks what happens and in week three contacts only the 2,560 people whose second week emails were correct.  And so on.  By the end of week ten, he will have whittled it down to ten people who have had ten correct emails – and the chances of them signing up in week eleven are very good indeed.

Now my brain hurts and I am going to have a biscuit break.