Saturday, 21 May 2016

Why The Unthinkable Should Happen Even Less Often Than You Think


At the time of writing we are currently half way through the 2016 FA Cup final - which got me thinking: what a tremendous season the Premier League has been, not just for sport, but also for anyone that is interested in probability and enjoys those startlingly rare and unpredictable outcomes in life. Against all odds (5000/1 at the start of the season) Leicester City have done the unthinkable - they have won the Premier League, finishing ahead of much bigger teams like Man City, Arsenal, Man Utd, Tottenham, Liverpool, and last season's champions Chelsea - all teams with much more money than Leicester and for the most part better players (consider how many Leicester players would get in those teams, and it won't be many).

We mustn't underestimate just how startling this is - it's not just the most shocking upset that's ever happened in the history of football, it's one of the most 'against the odds' things that has ever happened in British history full stop. What compounds the shock factor is that while you expect upsets in cup competitions, the league consists of 38 games where the best team that season has the best chance of being champions.

Perhaps there is another reason, though, why an upset of this magnitude was eventually likely to happen. It's noteworthy to me that there is a marked difference between team sports (like Football and Rugby) and individual sports (like Tennis and Snooker) in terms of how often the superior participants beat the inferior. In team sports the better teams manage to lose against the worse teams much more frequently than in individual sports, where the better players tend to win far more often.

I think I can work out why this is the case; it's almost certainly to do with probability and ratios in relation to consistency. Given that a team's performance depends on multiple players, teamwork, group cohesion, communication, and so forth, it is understandable that their rates of consistency are slightly less reliable than individual performers.

But in sports with individual participants there is another thing that increases the probability of the best player winning most often, it's to do with numbers, and a little thing called binomial distribution. Consider normal distributions in coin tossing - with a fair coin you expect to win about half the time. In a first to ten competition a fair coin would confer no advantage on either player. But suppose the coin was biased to represent superior ability in an individual sport, let's say tennis. Say you have a 1 in 4 chance of winning a coin toss with a biased coin, what chance would you have of beating your opponent in a first to ten competition? Using combinatorial methods that I won't bore you with, I calculate that your chance of wining a first to ten is less than 1%.

Obviously a competition involving a coin that gives you only a 1 in 4 chance of winning a single toss is going to be a harder competition for you to win the more coin tosses involved. That is, you've got less chance of winning a first to 25 than you have a first to 10. Your best chance of winning would be if the competition was a single coin flip - then you have a 1 in 4 chance. Any increase in coin tosses decreases your chances of winning. In a first to 2 your chances are 1 in 16 (25% x 25%), and in a first to 3 your chances are 1 in 64 (25% x 25% x 25%), and so on.

While sport is not quite the same as coin tossing, the same kinds of principles apply. If you as an amateur were told you had to beat a tennis player, or snooker player, or pool player hugely superior to you in ability, your best chance of beating them would be in a one game competition. The greater the increase in number of games you need to win (first to 3, first 5, etc) the less chance you have of winning, because, fairly obviously, there is more opportunity for the superior player's superior skills to affect the outcome.

What all this shows is that once the numbers are stacked against you (as they are when you play someone at sport who is better than you), those numbers get even more stacked against as the game goes on. For example, let me ask a question and see what your intuition says. Suppose you have a big tennis match against Andy Murray, with a £1 million stake, and suppose you take a magic pill that that makes you good enough to win 40% of the points (in Tennis every game is essentially a first to 4 - that is, 15,30,40, game). What are the chances that you'll win the match (best of 5 sets)  against Murray? 40%? No, way out. 20%?, Nope, still way off. 10%? No. 5%? Still not that close.

Your chance of beating Murray is actually just slightly less than 0.0005% (that's 1/20 of 1%). Think about it, if you are able to win 40% of the points against Murray you would win a game just 26% of the time (that factors in the times you'd need to win the game by 2 clear points). To win a set you'd need to win 6 games, which reduces your chances to just under 5%. As you have to win 3 sets to win the match that's why you have only about a one twentieth of 1% chance of beating Murray in a match.

Given the foregoing, and that such narrow probabilities should, on paper, apply to a team like Leicester City too, it is all the more remarkable that they went on to win the Premier League. Enjoy it while it last though, for I doubt anything this remarkable will happen again for a very long time.

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