# Probability is not intuitive

I have a (bad?) habit where I leave comments on blogs, arguing that the blogger has demonstrated a flawed understanding of statistics and probability.

Before I started doing this, I expected that my comments would be ignored. Instead, I almost always get responses. And generally speaking, the responders accuse me of being an idiot who either flunked out of Statistics 101 or never opened a Stats textbook in my life.

I don’t think any of those statements describe me. But rather than credential dropping, I usually try to win the argument substantively. This strategy has its limits, because statistics and probability just aren’t intuitive.

Take Jensen’s Inequality, which tells us that E(1/X) does NOT equal 1/E(X).

E(X) is the “expected value of X,” often called the mean of X. So Jensen’s Inequality says that if we are going to make a draw from the height distribution of a high school, the expected value of “1 over the drawn height” does not equal “1 over the expected value of the drawn height.” (Intuitive explanation: we are making draws from different distributions.)

Jensen’s Inequality also tells us that E(X/Y) does NOT equal E(X)/E(Y). If you think about this for a second, you can see why it makes predictions regarding exchange rates more complex than you might think. And if you think about it a little longer, you realize you could easily devise a casino game that would exploit the counter-intuitiveness of Jensen’s Inequality and make you rich.

But if some internet commenter had told me that E(X/Y) does not equal E(X)/E(Y) before I saw the proof for Jensen’s Inequality, I don’t think I would have believed them. Furthermore, I bet I make hundreds of assertions each day that my stats professors would find hopelessly naive. So I’m neither chastising nor upset by those people who call me an idiot when I talk about Simpson’s Paradox, the Prosecutor’s Fallacy, selecting on the dependent variable*, the ecological fallacy, or a million other things.

I am not sure what the moral of this story is. While I wish people had more exposure to statistical and probabilistic reasoning, the people who call me an idiot are precisely those people who have taken Stat 101, but nothing more.

I always knew that having an advanced degree in sociology would not make people any more likely to believe what I had to say about social processes if it went against their intuition. I guess I am just shocked that advanced degrees in statistics aren’t worth much more.

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* Selecting on the Dependent Variable: My favorite example of this is the “Speak of the Devil” phenomenon, wherein people conclude “ESP exists!” because many times in their lives, they have thought of a friend only seconds before that friend calls them.

Basically, I tell these people that you need to weigh the few times where this phenomenon did occur against every time you have ever thought of someone who then DIDN’T call you in the next ten seconds. And once you realize how enormous this number is, then you realize how likely it is that the “Speak of the Devil” phenomenon will occur multiple times in your life just by chance.

This fallacy involves mistaking the likelihood that a given event will occur (Bob calling me after I mentioned Bob on February 22nd at 3:40 pm) with the likelihood that any instance of a more general phenomenon will occur (Someone calling me after I have thought them).