A conversation with Fat Tony and Dr. John

In his popular book The Black Swan, Nassim Nicholas Taleb provides an interesting alternative to modern statistical theory. He theorizes that the world is driven by singular events that are highly improbable and bear an extreme impact. Along the way, Taleb heavily bashes statistics, essentially calling it an irrelevant, concocted pseudoscience (he especially enjoys haranguing about the fallacies of the bell curve).

In one fictional example, Taleb tries to show why statistics is so irrelevant to the real world by facilitating a conversation between a street-wise Brooklyner, Fat Tony, and the prototypical nerd/stats star, Dr. John. Let’s have a look (Ch. 4 The Ludic Fallacy, or the Uncertainty of the Nerd):

NNT (that is, me): Assume that a coin is fair, i.e., has an equal probability of coming up heads or tails when flipped. I flip it nineny-nine times and get heads each time. What are the odds of my getting tails on my next throw?

Dr. John: Trivial question. One half, of course, since you are assuming 50 percent odds for each and independence between draws.

NNT: What do you say, Tony?

Fat Tony:  I’d say no more than 1 percent, of course.

NNT: Why so? I gave you the initial assumption of a fair coin, meaning that it was 50 percent either way.

Fat Tony: You are either full of crap or a pure sucker to buy into that “50 pehcent” business. The coin gotta be loaded. It can’t be a fair game. (Translation : It is far more likely that your assumptions about the fairness are wrong than the coin delivering the ninety-nine heads in ninety-nine throws.)

And then Taleb proceeds to bash statisticians and their “inside the box” thinking. Now I hate to do this, but let’s apply some simple stats to this problem. Anyone who has take any kind of stats class should be able to follow along (although most won’t remember anything from stats beyond feeling their eyes becoming heavier and drifting off slowly into sweet…).

Let’s test the hypothesis that the coin is fair (p=0.5) given that the last 99 flips produced heads every time.

We have:

H0: P = 0.5              n = 99                    pˆ = 1 (99 heads for the last 99 flips)

Let’s decide to reject H0 if the difference between our sample p (pˆ) and the p according to our hypothesis is “significant” with a 1% significance level. In other words, if we are less than 99% sure that the difference between what we sampled and what the hypothesis we are testing holds is not due to random sampling error, then we cannot say that our hypothesis that the coin is fair is wrong.

Using the test statistic (a test we can use since what we’re looking at looks like a bell curve) we can determine the distance from what we sampled to what we think is the right number in a general unit, the standard error.

Basically, what we want to know is whether the number we got, 1, is far enough from 0.5, the original condition, for us to question if the original condition is true. Since we are trying to prove with 99% accuracy that the original condition isn’t true, we are looking for a value of z that is greater than 2.575. In other words, if the sample we got (1) is 2.575 standard errors (an arbitrary generalized unit) away from .5 then we can say with 99% accuracy that the coin is not fair.

Pluggin’ in some values, we get a z value equal to around 9.95. Way, way, way over 2.575. We can most definitely say with at least 99% accuracy that the coin is not fair.

So let’s go back to Dr. John. If Dr. John was an actual statistician, instead of some dude with a medical degree (or a PhD in psychology, I don’t know), he would have clearly rejected the initial condition of the test, based on the fact that if it were true, he would be getting results much closer to the initial condition (around 50/50 heads and tails). So statistics isn’t as useless as you thought. And Taleb isn’t as right as he thought. Taleb certainly makes very interesting points about Black Swans (it’s true, stats has a hard time accounting for them) and the normal distribution, but we mustn’t forget to be as skeptical of Taleb as he is of statistics. One of Taleb’s main arguments throughout the book is to be skeptical and keep your mind open to unforeseen possibilities.

If you haven’t read The Black Swan, I would highly recommend it. Of course, as Taleb might say, don’t believe everything you see on paper.