Faux Lucky Numbers

I came home from work tonight, sat down with my tuna steak and veg, switched on the TV and watched Deal Or No Deal.  It’s not a terribly good show, but tonight something interesting happened.  First, here’s a description of the game (from Wiki):

The game features a single contestant trying to beat the Banker, as they open twenty-two identical sealed red boxes assigned to potential contestants in an order of their choosing. The boxes contain randomly assigned sums of money inside ranging from 1p to £250,000. The day's contestant is selected at the beginning, bringing their box to the chair. As the boxes are opened over a number of rounds, the Banker makes offers of real money to gain possession of their box. The gameplay is coordinated by Edmonds, who communicates with the unseen banker by telephone. Contestants can either 'deal' to take the money, or play to the end, settling on the amount in their box.

What was special about tonight’s episode was this; the contestant’s godson had drawn a premonitory picture beforehand, depicting box number 22 containing the sum of £20,000.  After 20 of the boxes were opened, the contestant, remarkably, was left with box number 22 (which he brought to the table after a random 1/22 choice) and box number 13. I say ‘remarkably’ – but just how remarkable is it?  Well the boxes are chosen at random – the contestant happened to choose box 22 (as his godson predicted), and there are 22 sums of money – so the exact odds are 1 in 484 (1/22 x 1/22). 

Naturally everyone in the studio was getting excited; here sat a picture of box 22 with £20,000 in it, and here sat our contestant who coincidentally picked 22 at random, and who still had the figure £20,000 left in play (along with £5).  As you’d expect he chose to ‘No Deal’ when offered £6750 by the Banker, and opened his box 22, ready to be met with £20,000.  But, much to the audience’s astonishment, no £20,000 – the box contained the £5.  The premonition wasn’t really a premonition at all – it was just one of those coincidences of which they should have been mindful beforehand (and perhaps were). 

Of course, the explanation is simple – it’s the old ‘law of large numbers’ phenomenon.  Basically, even though the odds of it happening are a slim 1 in 484, there are so many games played that coincidences like this are bound to pop up every now and then. As pattern seekers we humans distort the real picture by editing out all the extraneous information and focusing only on what we see to be radical breaks from normalcy.  The old ‘These things always happen in threes’ fallacy is a good example – when people construct this illusory triune pattern, they do so only by forgetting all the times that things did not happen in threes. 

From watching this show a few times a month, and observing the many times contestants perceive as patterns or lucky legacies what is really only inevitability in the law of large numbers, it’s quite clear that most contestants (and by that I infer most people) do not understand this principle.  I don’t think host Noel Edmonds understands it very well either – you will often hear him say things like “£250,000” came to the table yesterday – what are the chances it will come twice in a row?” as though that somehow lessens its chance of returning the next day as well.  Guess what Noel, the odds are still 1/22, just as they were yesterday.  If you play the games long enough you’ll find all sorts of interesting patterns emerge, and that is the law of large numbers at work - but the odds never change at the level of the particularity. 

A lot of contestants play the game emotionally, by making reference to their ‘lucky numbers’.  There are two kinds of perception of lucky number, and they are related - the ones in prospect and the ones in retrospect.  You only have the former because you have the latter, and both are illusions, but one is much more dangerous than the other.  The ones in retrospect are based on ideas about past patterns; for example, John might say his lucky number is 3, because he was born on 3^rd April, he met his wife on the 3^rd December, he bought 3 scratch cards when he won £10,000, and he has 3 children.  That might make him leave box number 3 until the end – it has always brought him luck, so let’s hope it continues. 

But, of course, the reality is, 3 is no more of a lucky box for him in prospect than any other number – it is only perceived that way through the lens of retrospection.  When it comes to these kinds of pattern, the lens of retrospection is an unreliable predictor of the future.  If for John box number 3 contains a big sum of money it will no doubt perpetuate the illusion of it being his lucky number – but this is a philosophy by which one should not be ensnared.  It’s all very well thinking fondly of a number because it is associated with fortunate dates or events, but runs of events mean nothing with regard to the prospective lucky numbers, just as a run of five reds on a roulette wheel does not mean next time out you’ve got more chance of landing black, and a run of seven heads in a row with a fair coin does not mean that your next coin toss is more likely to be tails. Superstitions and frivolous pattern-seeking both remain ok as a bit of fun – but don’t be fooled by these concepts – they are illusory and can lead you astray if you overlook the law of large numbers.