n the 2010 World Cup, Paul the Octopus (in a German aquarium) became famous for being correct in all eight of the predictions it made, including predicting Spain over Germany in a semi-final match. Before each game, two containers of food (mussels) were lowered into the octopus's tank. The containers were identical, except for country flags of the opposing teams, one on each container. Whichever container Paul opened was deemed his predicted winner.1 Does Paul have psychic powers? In other words, is an 8 for 8 record significantly better than just guessing?

For each game, there are only two possible outcomes. Either Paul guesses the winner correctly, or he does not. The probability of guessing the winner of a game is independent of other games. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

The expected value of the binomial distribution is:

[tex]E(X) = np[/tex]

The standard deviation of the binomial distribution is:

[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]

Outcomes that are more than 2.5 standard deviations from the mean are considered unusual.

Paul guesses one team out of two.

So [tex]p = 0.5[/tex]

Eight games

So [tex]n = 8[/tex]

1 Does Paul have psychic powers? In other words, is an 8 for 8 record significantly better than just guessing?

Let's find if 8 is an unusual outcomes.

[tex]E(X) = np = 8*0.5 = 4[/tex]

[tex]\sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{8*0.5*0.5} = 1.41[/tex]

Guessing 6 or more games correctly are unusual outcomes.

Guessing 8 games correctly is an unusual outcomes, significantly better than just guessing, which means tht Paul has psychic powers