Talking conditional expectation with the boys thanks to examples from Alex Kontorovich and Gil Kalai

I saw a neat twitter thread from Alex Kontorovich yesterday:

The concept of conditional expectation is (surprisingly?) difficult for students, until they do the following exercise:

Let X and Y be iid 6-sided dice rolls, and let Z=X+Y be the sum. What is E(X|Z)? Def: E(X|Z)=sum x * Prob(X=x|Z). Not so enlightening. But space of values … https://t.co/z8ogdmvnIG

— Alex Kontorovich (@AlexKontorovich) August 29, 2020

It reminded me of a fascinating conditional expectation problem on Gil Kalai’s blog from 3 years ago:

Here’s the original blog post where you can fill in your guess in the poll (and see the solution, too!):

https://gilkalai.wordpress.com/2017/09/07/tyi-30-expected-number-of-dice-throws/

I thought that talking through these problems would make a nice project for the boys today, so we started in on Alex’s problem. The nice thing right from the start is that the boys had different guesses at what the expected value of one die would be when the combined roll was 8:

Now that we had a good discussion of the case where the sum was 8, we looked at a few other cases to get a sense of whether or not the intuition we developed from that discussion was correct:

Next I introduced the problem on Gil Kalai’s blog – again the boys had different guesses for the answer:

I had the boys write computer programs off screen to see if we could find the answer to the problem on Kalai’s blog via simulation. The interesting thing was that the boys approached the problem in two different ways.

First, my younger son started looking at dice roll sequences and he stopped when he found a 6 and always started over when he saw an odd number. He found the expected length of the sequence of rolls was roughly 1.5:

My older son looked at dice roll sequences and he stopped when he found a 6 but instead of starting over when he found an odd number, he just ignored the odd number. He found the expected length of the sequences looking at them this way was 3:

This turned out to be a great project. I’m glad that the boys had different ideas that we got to talk through. These conditional expectation puzzles can be tricky and subtle, but they are always fun!