Image Credit: *Daily Mail Online*

If it’s the Psychic Network, why do they need a phone number?!

— Robin Williams

He probably saw it coming, but for fans of Paul the psychic octopus, the news of his passing yesterday came as a shock. He left in the night aged two and a half when all three of his hearts ceased to beat at the same time. The British-born 8-legged oracle from Weymouth, dazzled the world this summer at the World Cup in South Africa when he made 7 consecutive correct predictions in favour of Germany winning their matches in each round, when he selected mussels from boxes draped in the colours of teams about to meet on the pitch. He also correctly forecast who would win the final between Holland and Spain. After Spain’s World Cup victory, he became a hero in Iberia. Spanish businessmen even raised €35,000 as a “*transfer fee*” to have Paul as the main attraction at a gastronomic festival where he wasn’t of course on the menu.

The odds of him making a string of 8 correct predictions are 1 in 2 to the power 8, or 1 in 256. The fact that he did it anyway rocketed him into the hall of psychic fame alongside spoon-bending Uri Geller. His final prediction was that England would win the right to host the World Cup in 2018. Whether or not this will also prove to be true, only time will tell. But one thing is for sure, Paul has become a statistical star who is about to be immortalized in a range of clothing lines and mobile phone applications. Sadly, Paul was unmarried and left no larvae behind for those hedging their bets on upcoming events. He was a one-off wonder of the depths, loved worldwide.

Octopuses are special in many ways. The first are the ways they sense. Octopuses can distinguish the polarization of light, and attached to their brain are two special organs, called statocysts, that allow them to know the orientation of their bodies relative to horizontal. Their suction cups are covered in chemoreceptors that allow the octopus to taste what it is touching. They have 2 eyes, 3 hearts and 8 legs. Apart from the mathematically-obvious relation that 2 to the power 3=8, they are among the most intelligent of all invertebrates. Maze and problem-solving experiments have shown that they have both short and long-term memory. Due to their genetically-programmed death on reproduction, almost all of their behaviour experiencial rather than instincial. They have been trained to distinguish between different shapes and patterns and marine biologists have witnessed them at play – repeatedly releasing bottles or toys into a circular current in their aquariums and then catching them. Octopuses often escape even from supposedly secure tanks, due to their problem-solving skills, mobility and lack of rigid structure. They are known to have broken out of aquariums and survive for time in open air so as to reach another vessel containing food. They have boarded fishing boats, opening their holds to eat crabs. They are the only invertebrates that have demonstrated the use of tools. At least four specimens of the Veined Octopus (Amphioctopus marginatus) have been witnessed retrieving discarded coconut shells, manipulating them, and then reassembling them to use as shelter. Large octopuses have even been videoed catching and killing sharks. In some countries, octopuses are on the list of experimental animals on which surgery may not be performed without anesthesia. In the UK, cephalopods such as octopuses are regarded as honorary vertebrates under the Animals Act 1986, extending to them protections special to invertebrates.

But, what does statistics have to say about Paul the Octopus? And why are we turning away from random number generators and towards animal instincts when it comes to selection? In the run-up to the World Cup, an article on the internet entitled “Paul the Octopus hates symmetry” showed an amazing chart whose symmetry was breath-taking. Germany’s loss to Spain in Durban on the 7th of July put an end to it. Paul’s choice in favour of Spain broke the spell and his preference for Germany leaving us all on the edges of our seats.

Image Credit: *Blame it on the Voices*

Some might say, he always goes for the mussel that is closest to him, looks tastiest in the underwater glow, moves in the most mouth-watering way, or is simply closest to his preferred tentacle – that is if octopuses are left-or right-handed! A statisticians job is to make a list of all relevant variables (the time of day, the date, brightness, initial location and orientation, preference for mussel colour, reaction to colours and colour combinations in the flags appearing on the boxes as well as a potential multitude of other selection criteria that may be important to our eight legged friend) and then perform hypothesis testing. Interestingly, Paul the octopus took part in a blind experiment – literally. The species Octopus vulgaris is known to be almost certainly color blind – neither behavioral studies nor electroretinogram experiments have shown any discrimination of a color’s hue. While octupuses are still able to distinguish brightness as well as an object’s size, shape, and orientation and there is a prevailing view that they are drawn to horizontal lines, there were horizontal lines/shapes on flags that Paul chose as well as those he did not choose. Another potential bias may have been spatial preference or related factors such as light intensity. On six out of eight predictions for the World Cup he chose the right-hand box (from the camera point of view), skipping the boxes with the Australian and, later, English flag. On three out of seven predictions, the German flag was on the right-hand box, with it being the left-hand box for matches with Australia, Serbia, England and Spain – all correctly predicted by Paul, with Germany winning two and losing two.

If all potential sources of bias can be eliminated then, and only then can we can ask the question: is Paul’s psychic ability greater than that expected from chance alone? Could it be that out of 100 similarly-aged, Octopus Vulgari, then Paul is in the special minority whose performance put him in the positive tail of a Binomial Distribution?

Image Credit: *Michael Taylor*

Modelling the number of Paul’s successful predictions at this world cup as BI(p=0.5, n=8); that is, we have seight independent trials each having probability of success p=0.5 at each trial. In order to test whether Paul is psychic, we need to construct a 95% confidence interval for the probability of 4 successes (the null hypothesis expected from chance alone). A quick way to do this is to assume, using the Central Limit Theorem, that the distribution is approximately a Normal Distribution, calculate his z-score and test it against the confidence interval around the mean. Paul’s performance in the South Africa 2010 World Cup was 8/8. From chance alone with a choice of two options, we would have expected Paul to get 4/8 which is the middle of the distribution. The standard deviation is σ=1.414 meaning that his z-score for the population of 8 trials is z=(8-4)/1.414=2.8289. At the 95% level of confidence (meaning we would expect the statistics to be true 95% of the time), α=5% and the critical points of the standard normal distribution are z(α/2)=z(0.025)=1.9600. For the population parameters used here, the standard error is simply E=z(0.025)σ=2.7714. This means that chance alone would predict that Paul’s result would be in the range: {(4-2.7714):(4+2.7714)}={1.2286:6.7714}. But Paul got 8.000 meaning that we have to reject the null hypothesis that his result is the same as that expected by chance alone. In fact his result 8.000 really is statistically significant!

Perhaps the vast majority of Paul-clones of the future will make predictions no more significant than 50:50 chance. This is what we would expect if the prediction ability of octopuses follows a normal distribution. This of course assumes that they cannot perform better than chance on average. Even if this is true, Paul is the special one. He may not have beaten statistics but he is so special that, even if octopuses are used as randomizers of choice, he will be in the top 2.5% most “intelligent” octopuses ever to swim the depths. Just like a jackpot winning prize ticket fished out of the sea, Paul is one of nature’s delights. What a pleasure it was to witness him at work. Sit back, listen to the Paul the Octopus song by Parry Gripp and spare a thought for life in the tails of the distibution. This is where probabilities of things happening are lowest – but where the most interesting things happen.