Amongst the many tools I have to help me in my job is a large, well-thumbed book called the Directory of UK and Irish Book Publishers. Within its (increasingly tatty and tea-stained) pages are the contact details for pretty much every publisher and book distributor in the country, heavily annotated in my spidery handwriting. Today, I found myself needing to call a publisher to chase after some incredibly obscure book we’d ordered from them, and as I sat on hold waiting for the tiny customer services department to spring into action I idly scanned the other publishers listed on the same page. As my eyes alighted on “Troika Publishing” the tinny on-hold music changed – to, of all things, the “Troika (Sleigh Ride)” from Prokofiev’s Lieutenant Kije Suite.

How amazing! I thought. What are the chances of that very piece of music playing at the exact moment my eyes read the (not-terribly-common-in-English) word “Troika” on the page? One in a thousand? One in a million? I was tempted to explain this astonishing synchronicity to the harassed-sounding customer service girl who eventually picked up the phone, but she didn’t sound as though she would appreciate the significance. And, come to think of it, neither do I.

These particular publishers, as well as two other companies on the same page, are people I ring on, if not a daily basis, at least three or four times a week. I knew that Troika Publishing were listed on the same page, because I’d seen it, many times before. And in my lifetime, I have probably heard that particular bit of music hundreds of times – it’s on the radio every Christmas, for example. The coincidence of the two of them entering my consciousness simultaneously is mildly notable, but I must have read the phrase “Troika Publishing” a thousand times without musical accompaniment, and vice versa. Only when the two events occur together does the mind assign any importance to them.

This logic can be applied equally well to prayer, or complimentary therapies. Just because two things which occur regularly anyway happen to coincide does not imply any significant link between the two. When someone prays for rain, and it rains – lo and behold, a miracle! If it doesn’t rain, or if it rains when no prayers have been offered, the event is ignored. This particular strain of confirmation bias has its roots in a fundamental misconception of the workings of coincidence. To return to my experience this morning; given that I look at that page of my Directory several times a week (usually whilst on hold), and given that “Troika” is a popular piece of music for holding, it’s more statistically improbable for me not to have experienced this dual event at least once in my working life:

Assume that the publisher I’m calling has a thirty-minute CD of relaxing classical music on the go to keep customers happy while they wait. “Troika” is pretty short (about one minute thirty seconds), and I’m on hold for an average of about three minutes. The chances of my hearing “Troika” on any given call are therefore 0.005*, which means that about 1 call in every 200 will result in my hearing that particular piece of music. Now consider that I probably scan that page whilst listening to hold music about three times a week. Given that I work approximately fifty weeks per year, that means I’ll be looking at that page and listening to that hold music CD about 150 times per year. If I work here for a mere ten years, that means I’m likely to experience the same “Troika” phenomenon about seven times…**

In other words, if you pray for rain enough times, sooner or later you’ll need an umbrella.

*Maths time: probability of hearing “Troika” = t x c where t = (percentage of CD that is “Troika” (1.5 (minutes) / 30 (minutes)  = 0.005 = 5%)) and c = (percentage of CD I am likely to hear during my time on hold (3 / 30 = 0.1 = 10%)), so 5% (0.05) x 10% (0.1) = 0.005
**Maths again: number of times I’ll hear “Troika” = y x p x w where y = (number of times I’m on hold in a year), p = (percentage chance of hearing “Troika” on any given call) and w = (number of years I work), so 150 x 0.005 x 10 = 7.5