Mosteller's "Fifty Challenging Problems in Probability with Solutions" – Solving these gave me a very good grounding on how to think about probability. As with Newman's "A Problem Seminar", I suggest taking copious notes and to try generalizing your results.
My plan for this fall is to go through as many of the 50 challenging probability problems as we can. I don’t know how many are accessible to the boys, but hopefully most of them are with some help.
Today we tackled problem #1 – You have some red socks and some black socks in a drawer. When you pick two socks at random the probability of a red pair is 1/2. What is the smallest number of socks that could be in the drawer?
Here’s how I introduced the problem to the boys and their initial solution:
The second part of the problem asks what the minimum number of socks is assuming that the number of black socks is even. This problem gave the boys a bit more trouble, but was a great learning opportunity for them.
In the last video they showed that the number of black socks couldn’t be 2 or 4 with direct computation. Here they showed that it could be 6 using algebra. This was a nice opportunity for some factoring practice.
Finally, we went to the computer and wrote a short (and obviously inefficient) program to test out some solutions. It was fun to find a few more (sadly we were super rushed for time here, but still had a nice conversation).
I saw a fascinating tweet from Nassim Taleb last night:
The Fundamental Tail Asymmetry: an illustration using the drop in MERVAL, of how you can accept, never really reject fat tails unless you have some physical law bounding deviations. Adding to the book. [It is Popperian falsification applied to inference] pic.twitter.com/9vB1fTYrjL
It reminded me of this lecture Taleb posted a few years ago (particularly the part starting around 8:45):
So, after seeing the tweet last night I decided to take a shot at sharing some of Taleb’s ideas with my kids. The obvious problem is that the details are pretty advanced. The point I thought I could communicate, though, was the idea of (to borrow Taleb’s terms) Mediocrastan vs. Extemistan – the worlds where one observation shouldn’t change things that much vs one in which one observation can change your world view completely. After talking through those ideas, I thought it would be fun to show the boys how different some probability distributions from Mediocrastan and Extemistan look.
We started by talking about distributions that stay close to the mean (like height) and ones where one observation can be far from the mean (like wealth or damage from volcanic eruptions).
Now we took a close look at a bunch of random draws from a normal distribution. The idea here (and in the following two videos) is a high level introduction to the “probabilistic veil” concept from Taleb’s tweet:
Next we moved to the Cauchy distribution. I was hoping that the kids would be able to see that the draws from this distribution were so different from the draws from the normal distribution that they could say with some certainty that these draws here definitely did not come from the normal distribution. It was fun to hear they take some guesses at the max and mins as I increased the number of data points.
Finally, we looked at the class of stable distributions. I didn’t try to describe these distributions in any detail, but rather just said that there was a parameter that we could vary between 1 and 2. Here the goal was to see if we could say if a distribution looked like a normal distribution or the Cauchy distribution. We were also able to see that varying the parameter changed the distribution, but that it would be pretty difficult to tell from the data if the distribution came from the parameter being 1.2 or 1.5 – this is one of the points in the Taleb video lecture from above.
I’m reasonably happy with how this discussion went today – these are pretty advanced ideas to be sharing with kids. Fortunately Mathematica makes it somewhat easy to see how different the various distributions look. Hopefully this conversation helps the boys get a little peek at the ideas of probability distributions and also helps them understand that the process of going from data to a probability distribution can be extremely difficult (even with millions of data points).
After seeing this puzzle my plan for the morning was to share it with the boys, see what they had to say, and then see what ideas they had for solving it.
Here’s their initial reaction. Their first idea was to try a few different configurations to test out the idea that the configurations didn’t change the area:
Now they solved for the area when the two squares had the same size:
The next idea they pursued fascinated me – they wanted to solve the puzzle using the assumption that the big square had twice the side length of the smaller square. Eventually this idea is going to lead to a big surprise!
It took a minute to get going with the algebra, but then they began to make progress. Seeing this progress happen live is why I love working through math problems with my kids
Sorry this video ends so abruptly – the memory card in the camera filled up.
Downloading the movies and clearing the memory card gave the boys a few minutes to think a bit more about the problem. When we restarted filming they had a plan. The algebra work was a little tricky for my younger son, I think, but we made it through and showed that the total area of the squares in this configuration was the same as the area in the last one.
They realized that their solution wasn’t a full solution to the problem, but I’m really happy with the work they did. After we finished with this last video I showed them the full solution off camera.
I think online learning has a lot of potential and even tried to put together a calculus lecture video library way back in graduate school. So I hope this one has success.
One thing that caught my eye browsing through the course information was the course’s pre-test. My younger son has been studying algebra this summer and the pre-test seemed like it might be a good challenge for him.
It is 10 questions – I think the work below is a nice example of how a kid thinks through ideas in algebra. Here are the questions and his work on them:
— Dr. Chanda Prescod-Weinstein 🙅🏽♀️ 🇧🇧🏳️🌈 (@IBJIYONGI) August 9, 2019
After seeing the first tweet from Kayley Brauer I was hoping to find a way to talk about this new result with the boys, but didn’t really know what to do. Thanks to the tweet from Dr. Chanda Prescod-Weinstein, I learned that the LA Times had put together a terrific presentation that was accessible to kids.
First, I had my older son read the article on his own and then we talked through some of the ideas he had after reading it.
I thought that reading the article on his own would be a little too difficult for my younger son (he’s about to start 8th grade) so instead of having him read it on his own, we went through it together:
Obviously I’m not within 1 billion miles of being an expert on anything related to this new image of the Milky Way, but it was still really fun to talk about it with the boys. I’m very happy that advanced science projects like this one are being shared in ways that kids can see and experience.
Last week Numberphile put out a fantastic video featuring Neil Sloane:
For today’s project we explored the sequence described in the first half of the video. Namely, the sequence that begins with and then continues with depending on the greatest common divisor of and . See either the Numberphile video or the first video below for the full formula.
To introduce the boys to the sequence, I had them calculate the first 10 or so terms by hand:
Next we wrote (off camera) a Mathematica program to calculate many terms of the sequence, and studied what the graph of those terms looked like:
Finally, I asked the boys to watch the Numberphile video and the describe what they learned. They were both able to give a nice explanation of why the sequence eventually repeated:
I love math projects that allow kids to play with really interesting math and also sneak in some k-12 math practice. The first sequence in the Numberphile video is a perfect example of this kind of project!
I thought it would be a really fun problem to talk through with the boys – especially since kids can definitely say something about the n = 2 and n = 3 case.
Here’s the introduction to the problem:
After the introduction we talked about the n = 2 case.
Now we moved on to the n = 3 case. The boys had an interesting idea on this one that caught me a little off guard. The neat thing is that they were able to come up with a pretty good hand waving argument that in the general case the three runners would eventually form an equilateral triangle.
After that neat hand waving argument from the last video, we tried to find a more precise argument for solving the n = 3 case. Solving it in general was just a bit out of reach, but they did find an argument for why at least one runner would become lonely.
Since we ended up pretty close to the proof of the general case for n = 3, I explained the last step after we finished with the last video. I think this is a really nice problem for kids to play around with and I think that lots of young kids will find the ideas in this problem to be really fascinating.