Yesterday Nassim Taleb shared a short paper looking at the tails of various probability distributions.

What you see and what you don't see:The hidden moments for a probability distribution. What we called shadow mean in paper with @DrCirillopic.twitter.com/rnRBU9YwKn

The paper is not for kids – the math is advanced – but I thought there might be a way to connect some of Taleb’s ideas with the project that we did on the coupon collector problem yesterday. By coincidence, in that project we’d spent some time talking about maximum values in a bunch of repeated trials.

So, we started by talking about the distributions we saw in yesterday’s project – especially the distribution of the number of trials required to find all 5 coupons (or, maybe even more simply – the number of rolls required to see all 5 numbers on a 5-sided die).

Also in this video I’m trying to introduce the idea that Taleb was studying – can we say anything about the tail of a distribution having seen only 100, 200, or even 1,000 samples?

Now we moved to the computer and looked more carefully at our 100, 200, and 1000 sample trials versus a 1 million sample trial. The boys were able to see at a high level how the amount of unseen area in the tail declines roughly like 1/n where n is the number of trials. This was one of the results in Taleb’s paper that I thought they would be able to understand visually.

Now I switched to a distribution that you really can’t say much about even if you have millions of samples. The problem is the so-called “archer” problem that we’ve explored before:

First I introduced the problem and let my younger son notice that we were really studying the distribution of since he’s learning trig now.

Finally, we returned to the computer to see the strange distributions that come from the archer problem. Even thought the boys had seen some of the ideas here before they were still surprised. Try to guess some of the numbers along with them as you watch the video!