Paul the Lucky Octopus

According to the news, there’s a psychic octopus in Germany. Apparently, Paul the Octopus is able to predict the outcome of the FIFA world cup games. Well, they don’t actually know the octopus is psychic – maybe he’s just very good at analysing the strengths and weaknesses of the various teams. Or maybe he’s just lucky.

There’s some interesting math behind this story. Here are the relevant facts.

• Paul’s owners put some perspex boxes into Paul’s tank. Each box contains a tasty octopus snack (a mussel, in fact)
• Each box is decorated with the flag of a team. One box with the German flag, one box with the flag of Germany’s opponent in an upcoming match
• Paul eventually chooses one of the boxes.
• Paul’s owner deems this to be Paul’s “prediction” of the outcome of the match.
• Paul has correctly predicted the outcome of 6 out of Germany’s 6 world cup games so far.

of course, nobody asked Paul why he chose that particular box. Perhaps he thought that team would lose. Or perhaps he just likes boxes decorated with lots of yellow and red – if that’s what’s happening, I predict that Paul will tip Germany to beat Uruguay for the third place match (irrespective of who actually ends up winning). Are octopuses colorblind?

Anyway, Paul’s reasons are beside the point. However, they do illustrate a rather odd fact about probability.

Suppose I had set up an experiment like this, for example, with my son’s pet hermit crab. Ok, hermit crabs are not so active as octopuses, but never mind that. Suppose also, that hermit crabs have no clue about soccer football games, so his predictions are right exactly half the time. Well, he might get one right, maybe two. Three or four if he’s lucky, and if he’s very lucky, 5 or 6. Most likely, I’d have nothing special to write to the newspapers about.

But I’m not the only guy around with access to a hermit crab – or octopus or hamster or budgie or cat or dog or tarantula. There may be dozens – or hundreds – of pet owners who put their precious Pauls through a predictive procedure. Considering there are billions of pet owners around the world, it’s not too hard to imagine that 1300 of them try something like this, and capture the event on video.

About 650 of these will get the first game right. Not exactly 650, of course, but approximately. About 320 of these will get the second game right. After the third game, about 160 pet owners are getting a bit excited, and telling their friends.

Come the fourth game, and about 80 of these pet owners and their excited friends start to think that their pet is onto something. Maybe the local newspaper gets involved. Cameras are rolling, and there’s a big crowd watching when roughly 40 of these pets spoil everything by getting the 5th game wrong. On the other hand, another 40 get it right. Now everyone knows something special is happening, and 20 are not surprised when the 6th prediction is on track too. The other 20 are surprised – and very disappointed – when it’s wrong.

Paul is one of those 20 (or maybe 200 or 2 – we don’t really know how many pet owners have tried this experiment) who has been lucky enough to get 6 coin flips correct. Unfortunately, he now has the public eye on him, and he has a 50% chance of embarrassing himself over Germany vs Uruguay.

Well, embarrassing his owners actually. Paul wins no matter what – he still gets his tasty mussel. Of course, there are those letters threatening to turn him into calamari, but again, that’s his owners’ worry, not his.

The point is, Paul is no more likely to guess Germany’s last game correctly than I am¹. However, he also has a 50-50 shot at eternal internet fame. After all, Germany has only one game left in the world cup, and if Paul gets this right, he’ll be remembered always as the octopus who predicted every world cup game he tried. And since octopuses have a relatively short life span, he won’t be around to try his luck in 2014. It’s lucky for his owners he’s not a parrot.

However, despite Paul’s fame, and talk of him being “psychic”, and “able to predict”, he’d just be one of the inevitable 10 or so (out of 1200) who were lucky enough to guess 7 coin flips right. Extremely unlikely events do eventually happen, if you try often enough – and while Paul himself only tried once, I’d guess he’s only one of 1000’s.

Footnotes :
¹ I think Uruguay will win the third-place match, by the way, even though I think Paul will choose Germany. Of course, I can always edit this post if I’m wrong… 🙂