I’ve been working in the UK for the last month (which partly explains my reduction in rate of writing), and I was fortunate to see Derren Brown perform live in Glasgow the other night. The show was great – a fantastic mix of his amazing skills, stories about his personal life (including coming out as a gay man, and how he was unpopular at school), and some good solid advice on scepticism and rationalism.
One of my favourite aspects of the show was the way he was able to explain how improbable events happen all the time, given enough opportunities. At one point, he asked everyone to stand up. Straight away, he asked all the men to sit down. Next, he told all the women not in the 25-35 age group to sit down. After a couple more such reductions, the people left standing were women aged between 25 and 35 who had a dog (or dogs) and at least two tattoos. If you were a 30 year old, female, dog owner with several tattoos, you might feel a little spooked at this point – like Derren personally selected you from the crowd. At this point, however, the lights came on, and everyone was amazed at how many people were still standing (well, everyone except the mathematicians).
But the lesson is simple. If there is a 1/100 chance that a typical person off the street is a woman aged 25-35 with a dog and a few tattoos, and if you have 2000 randomly selected people in a room, you’d expect around 20 of them to fit all the criteria (the audience was about 1700 strong, and there were a few dozen people left standing). But if you were one of those people, and you hadn’t thought about the numbers, you’d be feeling pretty special.
A while ago, I heard of another case that illustrates the same phenomenon, perhaps even better. Suppose you’re looking through your email inbox, and you come across a message from a self-described sports tipping expert. She tells you Team A will beat Team B in a match on the weekend. You promptly go to the next email and forget about it. Next week, you get an email from the same person saying that, this weekend, Team C will tie with Team D. You think back to the previous email and look up the scores and, sure enough, Team A did beat Team B as the expert predicted. So this time you watch the game between Team C and Team D, and it does indeed end in a draw. This same pattern happens for another three weeks – you receive a prediction before the game, and the expert’s tips always turn out to be right – five correct predictions in a row. By this time, you’re impressed. The next email you receive from the expert says:
“You’ll no doubt have noticed that I’ve now correctly predicted the outcomes of five matches in a row. I didn’t just pick the favourites, though – some of these results were big upsets, so you can see I am highly skilled in the art of sport prediction. If you had bet $100 on each of the matches, you would have made a profit of $1,000. I can continue to send you my predictions for the remainder of the season, but you will have to pay for my advice from now on. You can keep receiving my tips for a modest up-front fee of $200, payable to…”
So you send off your $200, and now you’re really excited about the money you’ll make. How lucky you are to have discovered, quite by accident, such a useful source of reliable information! So, you make a few bets, following the expert’s predictions for the next few matches. But suddenly, the predictions are not so great. You find you’re winning some and losing some, and not really doing any better than you would if you relied on your own (far from expert) judgement. So, what happened?
Here is what happened…
When you received your original email, another 1,000,000 people received exactly the same one. Another 1,000,000 received an email identical to yours except predicting that Team B would beat Team A. A further 1,000,000 were told that there would be a draw between Team A and Team B. After the match (in which Team A beat Team B), the second and third groups of 1,000,000 never heard from the “expert” again. Then, the first group of 1,000,000 were split into three groups. The first group were told that Team C would beat Team D, the second that D would beat C, and the third that the game would end in a draw. Since the game did end in a draw, the first and second groups never heard from the “expert” again. The same pattern was followed for five weeks. At the end of the first week, the original target of 3,000,000 was cut down to 1,000,000 who had received the first correct “prediction”. At the end of the second week, approximately 333,000 people had received two correct predictions in a row. After three weeks, 111,000 had received three correct predictions, then 37,000 after four weeks. Finally, after five weeks, more than 12,000 people had received five correct “predictions” in a row. And imagine now if just 1% of these people signed up and paid their $200. The “expert” would pocket about $25,000. And all with absolutely no skill whatsoever.
The moral of both stories should be clear. If some outcome has a 1 in X chance of happening, and a group far bigger than X has the opportunity to achieve the outcome, you can expect the outcome to occur several times. People win lotteries, and people recover from “incurable” illnesses. “Almost certainly not” does not mean “certainly not”. A one-in-a-billion day happens to around 7 people every single day.
Well, I was thinking about this as I was on the bus to the airport this afternoon, about to head back home. And as we drove past a church, I noticed a big sign saying Try Praying. The idea is obvious. If you’re a skeptic, why not just try praying? You’ve really got nothing to lose, right? If there really is no god, then obviously nothing will happen, and you’ll have lost nothing – just a few seconds wasted, praying to an imaginary deity. But if there really is a god, your prayer might result in something amazing – you might get that promotion you’ve always wanted, your mother might recover from her illness, you might restore your relationship with your estranged family – the possibilities are endless. (And then I suppose you’d start attending that church.)
But think about it. Imagine if every resident of Edinburgh prayed a very specific prayer. Suppose they all prayed for some event to occur, an event with, say, a 1% chance of happening. Well, the population of Edinburgh is about 500,000. So you’d expect a total of 5,000 unlikely events occurring (seemingly) as a direct result of prayer. That’s a lot of “unexplainable coincidences”! And if everyone prayed for several 1% events, then the list of “answered prayers” would be very long, indeed.
But nothing would have been proved about prayer at all. If the Edinburghians prayed to Zeus instead of (presumably) the Christian god, the same number of desired outcomes would have been achieved (still by chance). Same with Allah, or the Flying Spaghetti Monster. You can even pray to me if you want – tell me what happens!
But, to the person whose son learnt to walk again, despite being given a 1% chance of ever getting the use of his legs back, it can seem like something truly special has happened. Such people will most likely end up in the pews of the local church, telling their amazing story for the rest of their lives, and providing others with a great “reason” to believe in God. In the meantime, for every such lucky family, another 99 sons are still in the wheelchair. The trick didn’t work for you – you weren’t selected, and you can sit down, now.
So, next time something unlikely happens to you, don’t assume it must be for a reason – it probably wasn’t. Just remember that if enough people try and achieve something unlikely, someone will achieve it. It has to be someone, and it just happens to have been your turn this time.