On my previous post about Paul the octopus, a commenter asked a couple of questions which I thought merited a separate post to address. The first:
"There is of course a detail that you've (likely intentionally) overlooked, which is that the chance of winning a football match is not usually exactly 50-50. Given that Germany are one of the world's top teams they could be expected to win more matches than they lose.
Having done a bit of research, I've discovered that since Paul started making his predictions (at least for the public) at the start of Euro 2008, Germany have won 22 games, lost only seven and drawn four. Their win record thus stands at 66.7% over the last two years, which is probably a fairer representation of their chances of victory in a randomly determined match.
Can your analysis take account of this?"
This is an interesting question, and it boils down (as happens surprisingly often with probability) to a matter of perspective.
Suppose you have a friend called Peter who knows a bit about football. He's successfully predicted the results of the same six games that Paul has. Since Peter knows about football, he knows that the chance of Germany beating Australia (for example) was probably not exactly 50%. Does this matter?
Well, not really. The analysis we carried out last time was testing a specific hypothesis - that Paul was picking teams at random. This was our 'null' hypothesis, our default state of belief, if you will. Our 'alternative' hypothesis was that he has done better than you'd expect him to by chance alone. In testing this I claimed that Paul's chance of predicting the winner - if he's just picking at random - is 50/50. Crucially, this doesn't depend on the real chances of either outcome. This might not seem intuitive at first, but imagine Paul was picking the team after the game had happened - at this point the winner is known, so if he's picking at random he has a 50% chance of picking the right team. Since Paul's picking doesn't (we presume) interfere with the outcome of the game, if we're assuming he's picking 'blind' then it doesn't matter whether he chooses before or after the result is determined.
So what about Peter? We could test the same hypothesis and we would come to the same conclusion. The only difference is that we're not (as) impressed because we would expect him to be doing better than chance anyway - he has extra information to help make his decisions. Paul, meanwhile, is just an octopus, and so no-one would expect him to know anything (except possibly how to count to eight).
On a separate note, and with regards to the probabilities we've calculated telling us that Paul has some apparently incredible ability, it's worth stressing that that isn't what we've shown either. All we've done is shown that if Paul was picking at random (as - call me a sceptic - he probably was) he's just got quite lucky. This in itself isn't really that remarkable though - Paul was only brought to our attention after a string of successful predictions. There may well have been hundreds of other octopuses/coins/babies making similar predictions and getting them wrong, and we've just got to see the one who got them right. If you see a golfer hit a hole in one it seems remarkably improbable, but if you think about all the millions of shots that didn't go in, that single event occurring doesn't seem so incredible.
But anyway, on to the second question:
"Secondly, I would note that there are three possible results in most football matches (win, loss, draw) rather than two, although there seems to be no way for Paul to predict anything other than a win or loss. So far none of the matches he has made predictions for have resulted in a draw, but the possibility exists nonetheless. How does that affect the overall dataset?"
This is a good point (and one which I ignored previously for the sake of keeping things simple), and an interesting one to discuss.
As we've discussed, if our (null) hypothesis remains that Paul is picking at random, his probability of picking either team is just 0.5. However, since it's not certain that one of those teams will go on to win the game, his chance of picking the winner is actually going to be less than that. For instance, if 2 in 3 games end in one of the two teams winning, Paul then has a 1 in 3 chance of picking the team that wins, a 1 in 3 chance of picking the team that loses, and a 1 in 3 chance of there being no winning team to be picked at all.
What this amounts to is that Paul's chances of correct predictions are in fact even lower than those we'd already calculated, but unless you are willing to believe an octopus has been keeping an eye on the football pages of Bild, the chances are he's just very lucky.