# The Probability of a Psychic Octopus

I’m at dinner with a large group of graduate students and post-docs. It wasn’t part of a conference or anything, it was a group ski trip to Wisconsin and the restaurant was the “Wausau Mine Co.”, which looked like a Rainforest Cafe after fracking. Obviously not a serious scholarly event. Anyway, as we’re waiting for our fried cheese to arrive, the conversation turns to sports. And then soccer. And then the 2010 World Cup. And then Paul.

You remember Paul, he was world-renown as the most successful prognosticator of the World Cup. He successfully predicted the results of all of Germany’s games, and he correctly picked Spain as the winner in the final. Paul was the sensation of the year, with hundreds of articles written and TV reports made acclaiming his amazing soccer intuition. Also, he was an octopus. Paul lived in a tank in an aquarium in Oberhausen, Germany, and he made his picks by choosing between two boxes with mussels in them and a different colored flag on each. This method allowed him to correctly pick eight out of eight results, making him the most successful non-malevolent octopus in history.

Now the people I dined with that night are all incredibly smart, being graduate students after all. But sometimes incredibly smart people can appear dumb when it comes to matters of probability. So when the other people at the table started talking about how amazing Paul was and how he was able to make so many correct predictions, I became quite annoyed. “No! No! No!” I yelled, “Paul didn’t do anything special! It was all luck!” I then tried to explain why it was all chance, but my alcohol-induced state (as well as theirs) made it difficult to get my point across.

But this isn’t a blog post about why I have trouble making friends. Instead, I’m going to explain why Paul was not so much a psychic cephalopod and more the world’s luckiest living thing.

For starters, The 2010 World Cup was not Paul’s first attempt at prognosticating soccer results. During Euro 2008, Paul picked the results of all of Germany’s games. He picked Germany as the winner each time, but Die Mannschaft lost to Croatia in the group stage and Spain in the final. This left Paul with just a 4-2 record in picking games for the tournament, a perfectly reasonable and likely result for an entity picking things by chance.

For the 2010 World Cup, Paul did pick eight correct winners in all eight games (and didn’t pick Germany every time either, which angered many people in his home country after the team’s semifinal defeat). The probability of picking eight winners in eight tries by chance is 1 in 256, or 0.39%. However, the unlikeliness of the event happening by chance does not make Paul a psychic. The important thing to remember is that Paul was not the only animal picking results of World Cup games. There was Petty the pygmy hippopotamus, Leon the porcupine, and Anton the tamarin monkey. There was Pino the chimpanzee and Apfelsin the hog. There was Sayco the Dolphin and Dirty Harry the crocodile. There were fellow octopuses Pauline and Qingdao. There was the also quite well-known Mani the parakeet. There were likely hundreds more family pets making predictions throughout the World Cup in poorly-rendered YouTube videos. And there were the millions of human soccer experts, fans, and impartial observers making predictions for each and every game.

With all these people making guesses, the question is no longer “What is the likelihood of one of them picking every result correctly” but “What is the likelihood that any one of them picks every result correctly?” As it turns out, if you have, say, 50 animals picking (by chance, obviously) the results of eight World Cup games, the probability of one of them getting all eight right is 19.5%, well within the realm of possibility, and more so when you realize that this practice of prognosticating animals has been going on for more than just the most recent World Cup, but also past World Cups and Euro Cups and Super Bowls and national elections. At some point, one animal was going to get a lot of things right, and that animal was Paul. And of course, when an animal, human or otherwise, does get a lot of things right, just that animal becomes famous while everybody forgets all the animals that got things wrong. Mani the parakeet, for instance, picked all four World Cup semi-final games correctly, but erroneously picked Netherlands to win the final. So nobody’s talking about Mani now. And any animal that messed up an earlier result likely lost any chance of gaining attention immediately. So Paul got the attention, the adoration, the mussels, the careful wording of “allegedly psychic” from the BBC, and the criticism from Mahmoud Ahmadinejad for spreading “western propaganda and superstition.”

But attributing someone’s (or something’s) run of great luck to skill or other factors is very common. Every year during the NCAA Men’s Basketball Tournament in the United States, millions of people fill out brackets in office pools attempting to predict the winners of 63 games. And many years, there is somebody who gets every game right and receives attention for it, but given the huge number of participants—as well as the fact that predicting most games isn’t a 50-50 chance due to skill gaps between many teams—somebody getting everything right isn’t very surprising. Moving outside of sports, portfolio manager Bill Miller managed to beat the S&P 500 for fifteen years straight between 1991 and 2005 while working for Legg Mason. He was named the “Greatest Money Manager of the 1990s” and one of the “most influential people in investing” by various finance publications. But while media outlets proclaimed the probability of a streak like his by chance alone (the financial markets being far less predictable than sports) at 1 in 500,000 or even 1 in 2.2 billion, when you consider the large number of people who also started out managing funds and started out in the years both before and after 1990, you realize that those long odds are for the wrong question. In his book The Drunkard’s Walk (which everybody should read), mathematician Leonard Mlodinow calculates the probability of some person, somewhere, at some time beating the S&P 500 for 15 straight years at 75%. It turns out that nobody having a streak as long as Bill Miller’s is more of an anomaly than somebody having one.

The important thing to remember is that the world is incredibly random and many things are out of our control. We like to think that we have more control or insight than we do, especially in the worlds of sports and finance, but we don’t. And whenever the media likes to fawn over an animal for its incredibly psychic powers or an athlete for his ludicrous run of late-game heroics, more than likely it’s all down to dumb luck. I do believe that many people know this, however, and for them, paying attention to these and other stories is their way of laughing at how ridiculous it all is. Because with the way randomness works, it can get really ridiculous.

-Max Wasserman