My younger son is working through an introductory probability and stats book this school year. This week he came across a problem in the Bayes’ Theorem section that really gave him a lot of trouble. I was a little caught off guard by how much difficulty he was having with the problem (and how little my help was helping!) but then we got a great bit of luck when Julia Anker posted her solution to the problem:
First I used a Venn diagram with just A and B. Once I had that one figured out I used that and the rest of the info to figure out the overlapping C parts. pic.twitter.com/83MoPQpeNU
I had my son work through Anker’s solution yesterday and today we talked about the problem. Here’s the introduction to the problem and to his thoughts about Anker’s work:
At the end of the last video I had my son pick two of the regions in Anker’s diagram to see if we could verify the calculation. Here’s the calculation for the first region:
Here’s the calculation for the second region and an overall wrap up on the problem:
I’m extremely graful to Julia Anker for sharing her solution to the problem. It is an odd problem to have spent 5 days on for sure, but thanks to the extra help my son was really able to turn the corner and understand the problem. Math twitter is the best!
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!
The game is easy to explain. You roll a fair 6-side die N times, where N is any number you pick. You also choose an amount of money to bet – say X. If you never roll a 6 in your N rolls, you win 2^N times your money back. If any of your rolls are a 6, you get $0 back. Salmon’s questions are -> (1) How much money would you bet if you could play this game once, and (2) how many rolls would you select?
I thought this game would be fun to talk through with my older son. Here I explain the game and he talks about a few of the ideas he thinks will be important for answering Salmon’s questions. He has some interesting ideas about “high risk” and “low risk” strategies. We also talk through a few simple cases:
In the last video my son was calculating the probability of winning the game in N rolls by calculating the probability of not losing. That’s, unfortunately, a fairly complicated way to approach the problem so I wanted to talk a little more so he could see that a direct calculation of the probability of winning wasn’t actually too hard. We talked through that calculation here. We also find that if you roll 4 times you have roughly a 50/50 chance of winning the game.
Before we played the game he wanted to calculate the expected value for your winning in this game. Here we do that calculation and find the surprising answer. We then play the game. He decided to bet $100 and roll three times, and . . .
This was a fun problem to talk through with my son, and I’m excited to talk through it with my younger son tomorrow to see if he reaches a different conclusion. It had never occurred to me to talk through this or any version of the St. Petersburg Paradox with the boys before, so thanks to Felix Salmon for sharing this problem.
Today I wanted to extend that conversation to an example I first learned from a talk by Ole Peters.
I introduced the additive version of the game first and asked the boys what they thought would happen:
After the boys thought about the coin flipping game where you bet $100 each time we moved to the same game where you bet your entire net worth each time/. This one is a little harder to think through, but they boys has some good intuition:
Now we moved to looking at the two games in Mathematica. Here’s how the additive game plays out:
Finally the boys got to see the surprise in the multiplicative game – average wealth increases, but eventually any individual players ends up losing all of their money:
This game is really fun to think through and also a nice example to share with kids to illustrate additive and multiplicative games. Happy that yesterday’s detour into multiplicative processes led us to this conversation today!
I saw an interesting tweet earlier in the week from Atrin Assa:
@nntaleb demonstrating central limit theory by adding together increasing numbers of individual uniform distributions. By the time you add 3 uniform distributions, the resulting distribution looks like a normal distribution. Beautiful. #RWRIpic.twitter.com/7guT0c4dig
I thought the idea would make a neat project for the boys today, and it turned out to be even more interesting that I expected.
I started by just having the boys look at draws from a uniform distribution. You never really get a chance to go back and see ideas like this for the first time, so I’m always really interested to hear how kids describe what they are seeing:
Next we looked at the average of two draws from a uniform distribution. The boys had different thoughts about what this would look like before we saw the outcome – that was fun to hear. My younger son had some interesting intuition based on a dice game he’d played previously:
At the end of the last video my older son wanted to check what would happen if we looked at the geometric mean rather than the arithmetic mean. This idea wasn’t what I was planning to study, but it seemed like a great idea so we tried it out. They both had interesting guesses at what this new distribution would look like:
We wrapped up today looking at the arithmetic mean and geometric mean of three draws. With the arithmetic mean they started to see the normal distribution appearing. With the geometric mean they didn’t recognize the shape – my guess is that they’d never seen a distribution like this before:
This was a really fun project. Nassim’s intuition about how to explain important ideas from probably and statistics is incredible. I love working through his ideas with my kids.
Nassim’s problem really isn’t accessible to kids, but a slight variant is -> how many sequences of 250 coin flips are there where no run of heads or tails is longer than 2 flips?
I decided to go through that problem with the boys this morning. It was just at the right level to really challenge them, but still fit inside of a 30 min project.
We started by looking at Xi’s problem and they both had pretty good intuition for which sequence was which:
After the short introduction we started trying to figure out how to tackle the problem about sequences where the longest run was at most 2. After thinking of a few other ideas first, they decided to take a look at some shorter sequences to see if that would help us get some intuition for how many had runs that were no longer than 2:
Now we took a look at the sequences of length 4. Luckily there are only 16 different coin flip sequences of length four, so we could write them all out. The boys found that there were 10 sequences with runs no longer than 2. That led to an idea about how many would work in general:
Now we had a conjecture – there would be 16 sequences of length 5 that had runs no longer than 2 – so we tried to count those sequences directly to see if the conjecture was right:
Finally, we sketched a general proof of the conjecture (I’m intentionally being vague on what it is to not give it away). This part was also a little difficult for the boys, but they eventually saw the right pattern and that pattern led to the general proof:
This problem made for a really fun project this morning and Nassim’s problem led to some great twitter discussions that lasted all week. I was happy to be able to find a piece of Nassim’s problem that the boys could tackle.
First we talked about the proof that e is irrational. My younger son saw this idea as an exercise in the number theory book he’s working through right now. The proof is accessible to kids, though a bit more difficult than some of the other proofs of irrationality the boys seen before:
Next we moved to the idea in Nassim Taleb’s tweet. The idea that and are so close together is a really important idea from calculus and the general idea has many important applications:
Finally, we looked at the tweet from Sonia and discussed the simplified mathematical problem in the tweet and the surprising relationship to e:
I think these three ideas are fun ones for kids to see. The proof that e is irrational is something that I’m pretty sure I didn’t see until college, but is definitely accessible to kids. The other two ideas are really important ideas from calculus and probability and definitely worth exploring many times!
Jordan Ellenberg had a great article about corona virus testing in the NYT last week:
The idea of group testing is fascinating all by itself, but it also has some great math lessons for kids in it. I thought it would be fun to introduce some of those ideas to the boys this morning.
We started with a quick explanation of group testing and then looked at a simple case – a group of 16 with 1 person having the virus:
Now we looked at a slightly more complicated case – 100 people and 10 have the virus. It turned out to be a little more difficult to understand than I was expecting, but they made some great progress understanding the ideas as we talked through this case:
Next I had them read and study Ellenberg’s article for about 10 min. Here are their reactions and some of the ideas they thought were important.
It was really cool to hear their ideas about the article. This project help me understand that the group testing idea is harder to grasp than I realized. After a few examples, though, I think Ellenberg’s article was accessible to the boys and helped them understand how / why group testing could be an important step in dealing with the pandemic.