While idly surfing the web the other day - and, let's be honest, how often do we surf otherwise - I came across a summary of the most recent 450 Monday Lotto draws (note that the numbers from this site to which I refer in this blog might have changed by the time you view them since they're updated with each draw). For those of you unfamiliar with the NSW Monday Lotto format, 8 balls are chosen at random from 45 numbered balls with the first 6 deemed to comprise the "main" draw and the last two designated "supplementary" balls.
Given that arrangement you'd expect each ball to be drawn amongst the main balls, on average, about every 7 or 8 draws. What caught my eye then was that the ball numbered 30 had gone 37 draws without being vacuumed up as part of the main draw. That's nine months without gulping fresh air, which seemed pretty extraordinary to me. But, the questions is: how extraordinary?
Well, there's a 39 in 45 chance that a particular numbered ball won't be selected in the main draw in any given week, so the chances of that same numbered ball racking up 37 consecutive misses is (39/45)^37 (ie 39/45 raised to the 37th power), which is about 0.5% or, if you prefer, about a 200/1 event. That's a slim chance in anyone's assessment and certainly would be deemed statistically significant in most journal articles. So, do I have grounds for questioning the randomness of the Monday Lotto draw?
Absolutely not. The probability I just calculated applies only to the situation where the long-term unselected ball in question was pre-specified by me, but the situation that actually pertains is that I noticed that ball from amongst the 45 with the longest run of outs. I would have been equally amazed if it had been the ball numbered 5 that had achieved this record, or if it had been the ball numbered 12, or indeed any of the balls. In reality then I had a much greater chance of being amazed.
Just how much greater can readily be estimated via a quick simulation, which I've run and which tells me that I should expect to find at least one number with a run of outs of 37 weeks or longer about 20% of the time. In other words, it's only about a 4/1 shot and hardly worth being amazed about at all. Based on my calculations, we'd need to witness a run of about 47 weeks before we'd raise a statistician's eyebrow, as a run this long is the shortest that would surpass the 5% threshold for statistical significance. Further, we'd need a run of 58 weeks before we'd get that second eyebrow in motion as it'd only be then that we'd have a phenomenon with a probability of being due to chance under 1%.
Of course, events with probabilities even as low as 1% do occur occasionally - Melbourne to finish in the Top 4 anyone - so even if we did observe a run this long we couldn't definitively state that the Lotto draws hadn't been random, though we'd have a much stronger basis on which to suspect this.
My more general point here is how easy it is to be fooled into believing that something we've observed is extraordinary without realising how many non-extraordinary things we observed and discounted before registering the outlier. This psychological bias is the basis for many of the "unbelievable" coincidences credulously reported in the media - the person who wins the lottery for the 2nd time, the family with the Dad and the three kids all born on the same day, and the two holes in one by the same golfer in the same round.
If a lot of stuff's happening, most of it will be ordinary, but some of it must be extraordinary.
Matter of Stats
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