# Thoughts on Fooled by Randomness

Just finished Nassim Nicholas Taleb’s well-known book, Fooled by Randomness. Here are some brief thoughts, in no particular order.

#### The Birthday Irony

Despite the author’s years working in trading and writing a book on probability, in one of the few cases where he did actual math, he did it wrong. Here’s the original:

If you meet someone randomly, there is a one in 365.25 chance of your sharing their birthday, and a considerably smaller one of having the exact birthday of the same year.

Nassim Nicholas Taleb, Fooled by Randomness

It seems like he was trying to say – on average, there are $365.25$ days a year (first order approximation of leap years), so you have a $\frac{1}{365.25}$ chance of meeting someone of the same birthday.

If you do the math though, here’s the actual probability: every four years ($365 \times 4+1 = 1461$ days), there are $1460$ days in which your probability of sharing a birthday is $\frac{4}{1461}$, and $1$ day in which it is $\frac{1}{1461}$. So, the probability is $\frac{1460}{1461} \times \frac{4}{1461} + \frac{1}{1461} \times \frac{1}{1461} \approx \frac{1}{365.44}$. That’s significantly off from $365.25$ that you can’t really say “I just made a first order approximation”.

To fully understand this error, let’s say there is one extra day in $n$ years, instead of $4$. Then the number, instead of $365.25$ or $365.44$, will be $(\frac{365n^2}{(365n+1)^2} + \frac{1}{(365n+1)^2})^{-1}$. After taking Taylor series expansions, we get $365 + \frac{2}{n} - \frac{364}{365n^2} + O(n^{-3})$, or $365 + \frac{2}{n} + O(n^{-2})$, instead of the $365 + \frac{1}{n}$ that the author had guessed.

Let’s spend a little time to gain intuitions on why it’s $365 + \frac{2}{n}$ instead of $\frac{1}{n}$. Consider Alice and Bob, and a year is exactly $365$ days. Then the chance of sharing a birthday is $1$ in $365$. Now say we add $x$ days to Bob’s calendar only, so Bob’s birthday has $365+x$ possible choices while Alice still has $365$. Then, the probability that they have the same birthday is $1$ in $365+x$. At this point, it is clear that if we add $x$ days to Alice’s calendar, the chance of sharing a birthday goes down, therefore we know that the author’s estimate of probability is too high. Then, add $x$ to Alice’s calendar. If $x$ is small, we can ignore the probability that their shared birthday is on one of the days in $x$ (that probability is second order). Then, approximately we have the probability of sharing a birthday as $\frac{365}{(365+x)^2}$, which is close to $\frac{1}{365 + 2x}$, again ignoring the second order term. Substituting $x$ for $\frac{1}{n}$, we have arrived at the desired result. The factor $2$ comes from the fact that we added a leap day not only to Bob, but also to Alice.

Anyway, on a higher level, the lesson is that you should fully justify your simplifying assumptions, instead of jumping to conclusions.

#### Wittgenstein’s Ruler

This idea has never explicitly come to my mind, so I thought it was interesting. It says something like if you don’t have a reliable ruler, and you use it against a table, you might be measuring your ruler with the table. One example he mentioned was that some people in finance claimed that a ten sigma event happened. Using the principle – if you measured a ten sigma event, your ruler (mathematical model) is probably seriously flawed.

One takeaway from this is that statistics is merely a language to simplify and describe the real world, the world does not run according to the rules. It would be ridiculous to plot data points under a bell shape, and say that the world is wrong when the new data point doesn’t fit under it.

Another way of saying the same thing is conditional probability. Relevant xkcd: https://www.xkcd.com/1132/

One way I’ve seen it in real life is the current political situation in Hong Kong. Say there’s a certain probability that one citizen goes nuts and riot in the street, and there’s a certain probability that the government has done something terribly wrong. If you have very few people rioting, then the ruler tells you that those guys are probably at fault. But if you have a majority of citizens supporting the riots or rioting, then those guys become the ruler, and you’re measuring the government.

#### Think about All Possibilities

One very valid point in the book is that you should think of the world as taking one sample path in infinitely many possibilities. When you evaluate an outcome, you should think of all the things that could have happened. For example, if your friend did a thing and made a huge success, it doesn’t mean he made a good decision or that you should’ve done the same, or even that you should follow suit. We have only one data point, you don’t know what the probability distribution looks like. Maybe he could have lost it all. When you think about all that could have happened, you will have less jealousy to the lucky and more sympathy to the unfortunate.

#### Happiness is Relative

This is a tangential point to randomness, but still important to keep in mind. Given that you have basic human needs fulfilled, your happiness often doesn’t depend on how much you have, but how much more you have compared to those around you. More generally, it’s not the absolute well-being that matters, but the changes. So to be happy, don’t be the medium fish in the big pond, go to the small pond and be a king. If you start out at the top, tough luck, because chances are your status will revert to mean over time.

#### Limit Your Loss

If there’s one actionable item from the book, that’s to always remember to limit your worst case scenario. Between a steady increase in personal well-being with no risk of going bankrupt and more income but also a chance of losing everything, you should prefer the former, because eventually the unfortunate thing will happen. That’s called the ergodicity – any event with a nonzero probability will eventually happen, mathematically.

#### The Author’s Conspicuous Faults

I believe most readers will often find the author’s comments controversial and provocative, if not arrogant and overgeneralizing. There’s a bunch of stuff he said that is just plain wrong.

He said in the beginning of the book that he didn’t rewrite according to his editor’s suggestions, because he didn’t want to hide his personal shortcomings. But the point of a nonfiction book that is non-autobiographical is not to convey who you are, but to give readers inspirations and positive influence. If you say a bad thing in the book that you believe in, you’re not “being true”, you’re bad influence! I don’t know what exactly he was referring to, but I suspect they should include my following points.

He’s exceptionally arrogant, way off the charts. You’ll see him saying things like “I know nothing about this, despite having read a lot into it” and “I know nothing, but I am the person that knows the most about knowing nothing”. He just couldn’t write one sentence that ends in a defeated tone. Before he puts a period down, he must add another clause to the sentence to remind the readers that he’s just being humble, he didn’t mean it. It’s quite funny when you look for it.

He also loves stereotyping people to the extreme. He would say things like “journalists are born to be fooled by randomness”, “MBAs don’t know what they’re doing”, “company executives don’t have visible skills” and “economists don’t understand this whatever concept”. One thing he said in the beginning of the book was that he didn’t need data to back up his claims, because he’s only doing “thought experiments”. I think he mistook that for “unfounded personal opinions”. When you make claims about journalists and economists being dumb, that’s hardly a thought experiment. You absolutely need to back up your claims.

Overall, this book has some good ideas, but not that many. If you already have a decent background in math, maybe you can skip this book without harm.