Sunday, February 2, 2014

Hypothetical Homework #1: The Amazing Eli

I'm in a very awkward stage right now - I'm neither teaching nor taking any classes. This hasn't happened to me in almost a decade. Many (and probably most) of you would say this is a happy time and I should just enjoy it. Instead I've felt absolutely lost. I'm not taking, grading, writing, or studying for any tests. So I've decided to start a new series on this blog I'm calling "Hypothetical Homework". It's basically the questions I wish I could either ask students or answering questions that others send me.

Here's the deal: I'll answer at least one question per week. I do not guarantee the questions will be answered correctly - in fact I suspect I'll get some very, very wrong - but discussing why I'm wrong can be part of the fun (that's what we call fun at least, right?) Submitted questions will take precedence over questions I write for myself, so please send questions to chad (AT) or ask them in the comments or on Twitter. Any topic is allowed, but chemistry and related disciplines will likely get picked first unless the questions are really awesome.

So, without further ado I present to you:

Hypothetical Homework #1

Eli, an orangutan at the Hogle Zoo has correctly predicted the Super Bowl winner 7 years in a row, the latest being the Seahawks victory over the Broncos. Is this statistically surprising? Why or why not?

The result is not surprising. Eli has correctly predicted the winner of Super Bowl from 2008-2014.1 The probability of him choosing the winner in any given year is 1/2. For Eli to pick the winner correctly 7 years in a row is a bit more surprising:

That is, there is a 0.78125% chance that Eli would pick the winner of the Super Bowl 7 times between the year 2008 and 2014. Or you could say that there is a 99.21875% chance that Eli would have gotten at least one of those years wrong:

So far it actually seems like we should be amazed at this ape's ability, and this is probably where most people would stop. But here's the deal: we also need to ask ourselves what makes Eli so special? Was he chosen from birth and raised with the expectation that from 2008 to 2014 he would correctly predict the Super Bowl Champions? Probably not. So the question we want to answer is, "What is the probability that a captive orangutan (of which there are ~350) could correctly predict the winner?" For that I pull out a special trick in statistics that I like to use: calculating the probability of the opposite outcome. In this case I want to find the probability that all 350 captive orangutans would predict the wrong team at least once:

There is only a 6.42% chance of all orangutans getting at least one pick wrong. This means that there is a 93.58% chance that at least one of the 350 captive orangutans would have picked the winner correctly from 2008-2014. A counter point may be that not all 350 captive orangutans were asked to predict the winner, but that doesn't really matter given the number of animals asked about any number of sporting event. And we haven't even begun to mention the probability of this happening during any other 7 year period (2007-2013, 2006-2012, etc). The point is you should not be surprised to hear an animal has predicted the future because it's all a confirmation bias: Only the animals that pick winners make the news.

[1] - Why didn't Eli pick his brother's team for the win? (It's a sport's joke, nerds. Peyton Manning, Quarterback for the Denver Broncos, has a brother named Eli who also plays football)