Fooled By Lumpiness

In a typical AFL game in 2012 the winning team registered about 30 scoring shots and the losing team about 20. On the assumption that the sequence of team scoring shots is random - so that, for example, the winning team's probability of registering the next scoring shot is always 60%, regardless of whether or not it was the team to score last - how likely is it, do you think, that we'd witness a run of 5 or more consecutive scoring shots by the winning team is such a game?

The answer, which I obtained by simulating 100,000 games with 30 scoring shots for one team and 20 for the other, distributed at random, is almost 90%. Even for the losing team there's a better than 20% chance that 5 or more of its 20 scoring shots will have occurred as an unbroken sequence. Both those results, I'd suggest, run counter to our intuitions.

Longer sequences can also crop up at random more often that we might reckon. For the winning team, a run of 8 or more successive scoring shots has a better than 20% chance of occurring, while for the losing team such a streak is very unlikely, but at just under 1%, not impossible.

Had we primed ourselves to look for evidence of "momentum" made manifest in scoring streaks, we'd be at risk of being fooled by that subset of these average games - which would be about one-in-five of them - where both the winning and the losing teams registered scoring streaks of length 5 or longer, despite the fact that both teams' scoring shot sequences were completely random.

Depending on the total number of scoring shots in a game and how these are distributed between the winning and losing teams, the relative likelihood of scoring streaks of the types discussed above will vary. The table below provides this information for five "archetypical" game types:

Note that we don't need for one team to be clearly superior to the other before we expect to see long streaks of scoring. In fact, the Evenly Matched-High Scoring game type, where both teams register 30 scoring shots in the game, is the one that seems most likely to convince us that scoring is streaky even when it's really random. In such a game there's over an 80% probability that at least one of the teams will embark on a 5-in-a-row scoring shot binge and a probability of over 40% that both teams will rattle off such a streak in the same game.

Randomness really is a lot more lumpy than we expect.