Today’s news sources are talking about Leicester City’s winning the Premier League as a sort of miracle. The bookies’ initially-offered odds of “5000 to 1” has morphed into a supposedly scientific/mathematical measure of probability — we are being told that Leicester City had “a slim chance of only 1 in 5000” of winning the Premier League. Yet amazingly, they did win it! We are given to believe that “1 in 5000” is a numerical measure of how surprised we should be at the fact that they did in fact win.
That is ridiculous. The Premier League consists of 20 teams, chosen specifically for their ability to beat other teams. Suppose instead of the Premier League on its own, we imagine a much larger competitive free-for-all containing the Premier League, plus the First Division below them, plus the Second Division below them, and so on, till we have 5000 teams altogether playing against each other.
If we knew nothing whatsoever about any given team, in that situation we might assign a “probability of only 1 in 5000” that it would win. In other words, if we picked a team randomly from the 5000, and did so repeatedly, then in the long run we would pick the winning team about once in every 5000 attempts to do so.
But now suppose we are told something about a given team: that it is in the top 20. That should make us raise our numerical assessment of its chances of winning the free-for-all. If we were further told that a team in the top 20 never loses to a team in the bottom 4980, we would very significantly raise our estimate of its chances of winning the free-for-all. It would be something similar to playing the Monty Hall game, except that instead of one out of three available doors being ruled out, 4980 out of 5000 available doors are ruled out.
But that, in effect, is what limiting the free-for-all to only the Premier League does. It means that if we know nothing at all about a team, the repeated act of picking one out randomly in the hope of choosing the winner would be successful much more often than 1 in 5000 times.
To lower the “chances of winning” in the face of further knowledge about a given team is to introduce capricious, subjective factors that cannot be relied on to make statistical judgements of relative frequency. They involve unrepeatable events or events that are not statistically lawlike, and so cannot be reliably extrapolated from. All we can do is guess about credibility here.
Casinos make money reliably because the behaviour of dice, cards, rotating cylinders etc. is statistically lawlike. For example, we know that in the long run about one sixth of rolls of pairs of dice will be doubles. But the behaviour of football teams in the Premier League is not at all lawlike. Bookies have to use numbers in their line of work, but let no one think these numbers correspond to measures of anything real or significant.
I suggest that we should sharply distinguish statistical relative frequency and subjective judgements of credibility. Numbers measure the former, but their presence is a will o’ the wisp when we are dealing with the latter.