Why everyone should have assumed Jian Ghomeshi was lying (courtesy of math)

You've probably seen some form of this critical thinking exercise. There's this description of a person. You're asked to choose their likely profession. The description usually looks something like this:

A person is neat and orderly. An introvert, the person is shy and does not easily make friends. In spite of near-sightedness, which requires him or her to wear glasses, the person also loves books and is an avid reader. Is this person a librarian or a teacher?

I probably don't need to ask what you picked: it was "librarian". Because that sounds like the stereotype of a librarian. If I had allowed you to attach percentages or probabilities, you might have allowed for the chance that this describes a teacher. You might have even said 50/50, which seems like a safe answer. And now that I've suggested it, you're probably thinking "that sounds reasonable", right? Right.

But, in fact, no. It isn't. You're wrong. You are so, so wrong.

That is to say, you're probably wrong. And you're probably wrong because you've likely paid attention to the wrong details. Rather than asking yourself "does this person sound like a librarian? or a teacher?" you needed to be asking "just how many people work in these professions, anyway?"

That being the real question, you probably already have some idea of the answer. According to the U.S. Department of Labor, there were about 15,000 librarians employed in that country in 2012. But teachers? Well, if we restrict the category to only people who teach JK through Grade 12 classes, there were about 3,000,000, give or take 100,000. That's about 200 teachers for every librarian.

So, for every shy, near-sighted librarian - and even if we assume that every librarian fits this description - do we really think that we can't find more people who fit the description among a random sample of 200 teachers? And not just more but even 10 or 20 or 100 more of them for every 1 librarian? Yep, there's at least one, maybe even two magnitudes more teachers who fit this description than librarians. There's no reason for hesitation, here. There's maybe a 1% chance that "librarian" was right. You should have guessed "teacher".

There's a name for this kind of reasoning: Bayesian inference. A Bayesian inference refers to decision-making based on probabilities. We know - or at least we should know - that there are many more teachers than there are librarians. Google then confirms this and allows us to attach a factor to it. Those details about being orderly and loving books? It's really neither here nor there, since it's a decision-making process that's based on a feeling or stereotype about what a librarian is like, rather on anything we can measure or prove.

Thus, our basic assumption - our a priori assumption, given that we know little else that is useful - should be that "teacher" is the more probable correct answer.

So, where is this blog heading? I'll cut to the chase.

Last week, I started to write a blog about the Jian Ghomeshi sex scandal. I'm not going to rehash the details. If you're reading this, you're probably familiar with the allegations against the former CBC radio host. If not, here's the Wikipedia summary.

At the time I started writing, three women had accused Ghomeshi of sexual abuse and assault and each of them had been interviewed by Jesse Brown, though none allowed for their quotes to be attributed. Another women that Brown interviewed accused Ghomeshi of sexually harassing her at work. And some of Ghomeshi's own friends - mostly famously Owen Pallett - said that they believe the accusers. Also, depending on where you live and whether you shared any mutual friends or friends-of-friends with Ghomeshi, there was plenty of rumor and innuendo that supported the accusers.

Still, there were no irrefutable "facts", as such, like videos or pictures. A lot of people on the internets made comments like these:

K*** M***: Nobody knows "the whole story" except for him and the women. I am not taking anyones side on this matter. 

@pothen: Why is anyone (aside from the parties involved & personal friends) mouthing off about #JianGhomeshi? We know *nothing* so far. 

Which seems reasonable, right? We don't "know" anything, do we? It's a 'he said'/'she said' and the allegations are as likely to be true as they are false. The probability is 50/50 and we just shouldn't choose sides.

If you've been following along this whole time, you can probably guess where I'm headed with this.

As a matter of fact, we do know something. We know that women almost never lie about sexual assault. And that knowledge should inform whether we think these women are telling the truth.

So, what exactly do we know. Here are a couple details that aren't perfectly matched to the Ghomeshi case, but indicate a pattern:

• We know that it's very rare for women to go to the police and falsely accuse a man of sexual violence. For instance, multiple researchers have put the rate of "unfounded" or false rape accusations between 1% and 8%.
• As well, the rate of "unsubstantiated" criminal domestic assault cases is estimated to be around 25-40%. "Unsubstantiated", in this case, adds all of the cases that were dismissed due to lack of evidence - not that the accusation was false or malicious, necessarily, though these would be included in the number, too.

Granted, these refer to court proceedings and not unnamed accusers in newspapers, but the trend seems clear - women tell the truth more often than not. Even the most pessimistic guess would have to start with the assumption that the likelihood an accusation is true is >50%.

But we're not dealing with an accusation, are we? We also know that there were three people alleging particularly heinous sexually violence, not just one. That increases the likelihood that he did these things to at least one of them by a significant margin: that >50% becomes something more like 90+%.

Let me connect this process more explicitly with my opening example. At first, you might have asked yourself "do I think Jian Ghomeshi did this?" And so you might have made a decision based, to some extent, on what you think of him. But that was the wrong question, and it might have led you to commit the same representativeness fallacy that you fell prey to, above, if you chose "librarian". A better question would have been something more like "someone has been accused of sexually assaulting three women - how likely is it that these women are telling the truth?"

So, even before you heard any of the details, you should have thought it at least 90% likely that the accusations were correct. That's your starting point: not 50/50, but 90/10. Because while we might not know the whole story - just as we didn't know the entire life story of the person who wears glasses and loves books - we know a lot about probabilities, and that has to inform our decision-making.

And that's before you factor in the other story of sexual harassment. The letter from Owen Pallett. The stories from former co-workers or whispers from women in bars. The other six women who would come forward to the Toronto Star in the next week, two of whom were willing to be identified by name. The internal communications from the CBC that complaints had been made. The letters from journalism schools that declared his show off-limits to students because of his past behavior. The recollections of university dons that they had to keep an eye out for him. The woman who recalled a creepy encounter with him on XOJane, last year. The woman who accused him of sexual assault via an anonymous Twitter account, this spring. Or the fact that believing the accusers requires that only Ghomeshi is lying, while believing Ghomeshi requires that a dozen women and at least one journalist have orchestrated an elaborate and long-running conspiracy...