Talking through a neat introductory probability / expected value problem from Pasquale Cirillo with my younger son.

This problem was posted by Pasquale Cirillo last week and I thought it would be great to talk through with my son:

This is the video where Cirillo talks through the problem if you want to see his full solution:

I’d asked my son to think about it a bit ahead of us sitting down to talk. Here are some of his ideas:

Here we find the expected winnings if we use my son’s first rule – stop in either of the first 2 rounds if you get a 6:



We ended the day today by talking through the strategy of stopping in either of the first 2 rounds if you see a 4, 5, or 6.



At the end of the last video I let my son know that we hadn’t quite found the best strategy, yet. Tomorrow we’ll finish up and find that best strategy.

The paradox of the 2nd ace part 2

Yesterday we did a project on this fun problem from Futility Closet:

Today we finished the project by talking about the 2nd part of the problem and then having a discussion about why the answers to the two questions were different. Unfortunately there were two camera goofs by me filming this project – forgetting to zoom out in part 1 and running out of memory in part 4 – but if you go through all 4 videos you’ll still get the main idea.

Here’s the introduction to the problem and my son’s solution to the 2nd part of the problem. Again, sorry for the poor camera work.

Next we went to the computer to verify that the calculations were correct – happily, we agreed with the answer given by Futility Closet.

In the last video my son was struggling to see why the answers to the two questions were so different. I’d written two simulations to show the difference. In this part we talked about the difference, but he was still confused.

Here we try to finish the conversation about the difference, and we did get most of the way to the end. Probably just needed 30 extra seconds of recording time 😦 But, at least my son was able to see why the answers to the two questions are different and the outputs from the simulations finally made sense to him.

So, not the best project from the technical side, but still a fun problem and a really interesting idea to talk through with kids.

Revisiting (again!) the Paradox of the Second Ace

Today my younger son and I are going back to a problem that we’ve looked at before – the “Paradox of the Second Ace.” This is a problem I learned about from Futility Closet:

Here’s the problem as described on their site

This problem teaches a couple of good counting lessons. Today we focused on the first part – if you have at least one ace, what is the probability that you have more than one. First, though, we talked through the problem to make sure my son understood it:



Next I asked my son to work through the calculation for the number of hands that have “at least one ace.” He made a pretty common error in that calculation, and we discussed why his calculation wasn’t quite correct:



Now we talked about how to correct the error from the last video via complementary counting:

Now that we had the number of hands that had at least one ace, we wanted to count the number of hands with more than one ace. My son was able to work through this complementary counting problem, which was really nice to see:



Finally, since we had all of our numbers written down as binomial coefficients and these numbers were going to be difficult to compute directly, we went to Mathematica for a final calculation:



Excited to continue this project tomorrow and hear my son’s explanation for the seeming paradox.

Part two of talking through the Bayes’ theorem chapter of How not to be Wrong with the boys

Yesterday we did an introductory project on Bayes’ Theorem inspired by chapter 10 of Jordan Ellenberg’s How not to be Wrong:

https://mikesmathpage.wordpress.com/2021/01/23/talking-through-a-bayes-theorem-problem/

Yesterday’s discussion helped the boys understand the problem that Ellenberg is discussion in chapter 10 of his book a bit better (hopefully anyway!). Today we took a crack at replicating the calculations in the book relating to the roulette wheel example.

First we revisited the example from the book to make sure we had a good handle on the problem:

Next we talked through the details of the process that we’ll have to follow to replicate the calculations that Ellenberg does. Following the discussion here the boys did the calculations off camera:

Here we talk through the numbers that the boys found off camera – happily we agreed with the numbers in the book.

At the end of this video I introduce a slight variation on the problem – instead of getting R, R, R, R, R in a test of 5 rolls, we get an alternating sequence of R and B for 20 rolls:

Here are their answers – and a discussion of why they think the answers make sense – for the new case I introduced in part 3 of the project:

This two project combination was really fun. My younger son said that he was confused by the roulette wheel example, but I think after these two projects he understands it. I think it is a challenging example for a 9th grader to understand, but with a little discussion it is an accessible example. It certainly makes for a nice way to share some introductory ideas about Bayesian inference.

Talking through a Bayes’ Theorem problem

My younger son is reading Jordan Ellenberg’s How not to be Wrong and the chapter talking about Bayes’ Theorem caught his attention this week. Looking around for something related to talk about in a project, I found this interesting problem on Wikipedia:

Before talking through that problem, though, we talked about the roulette wheel example from Ellenberg’s book:

Next we began to talk through the problem from Wikipedia. This part of the project shows the initial reaction and some thoughts on the problem from the boys:



Finally, with the initial thoughts out of the way we moved on to solving the problem. My older son was seeing these ideas cold, but what was really neat to me in this part is that the ideas from Ellenberg’s book really helped my younger son see how to solve this problem:



I feel like I got a bit lucky with this project. The ideas about updating probabilities looked a bit too difficult to go through in a 15 minute project – especially since my older son was seeing them for the first time. With this introduction, though, I think we can compute / verify the updated probabilities in the roulette wheel example from Ellenberg’s book in a project tomorrow.

Revisiting a neat coin flipping game I learned from Ole Peters

This morning I accidentally stumbled on an old coin flipping game we looked at last year:

I thought it would be fun to take a look at the problem again since the last look was long enough ago that the boys probably wouldn’t remember it. Here are their initial thoughts in the problem. After a bit of discussion, the boys came up with a good argument for why HHHT only would appear more often than HHHH only.



Next we looked through a simple computer program I wrote to model the situation. This isn’t the best or most clever way to write the program, but I thought it was an easy one to explain:



Finally, we looked at how the numbers would change if the sequence had 50 flips instead of 20. It was interesting to hear the boys explain why the numbers had changed – I think this extra discussion helped them understand the original problem a bit better:

Extending our project exploring the Pólya Urn

Yesterday my younger son and I looked at the Pólya Urn problem inspired by this sequence of tweets from Ole Peters and Marcos Carreira:



Today I had my son explore a little further. He was interested to see if different starting positions led to different distributions of endings. He looked at five different starting positions. Here’s the first (with a quick review of the problem) when the urn starts with 5 black and 5 white balls and we play the game 1,000 times:



Next he looked at how the starting position with 1 black ball and 5 white balls evolved. The way the distribution of the number of white balls at the end changes is pretty amazing:



Now for the most surprising one of all – the starting position with 1 white ball and 1 black ball – it seems that ending with 1 white ball or 1001 white balls (or any amount in between!) is equally likely:

Finally he looked at the starting position with 1 black ball and 10 white balls. This one is a little less surprsing having already seen the 1 black ball and 5 white ball game, but still it was neat to see:

This is a fun little game for kids to study. It is also a nice introductory programming exercise, too. Thanks so much to Ole and Marcos for sharing their ideas!


Playing around with Polya’s Urn thanks to Ole Peters and Marcos Carreira

I saw a great twitter thread earlier in the week:



With the results and the code presented so nicely, diving into Polya’s Urn with my younger so this weekend was a no brainer!

I started by explaining the problem and asking for his thoughts. His intuition about how the game and how it would play out was fascinating to hear:



After he explained what he thought would happen, we played the game once:



Next we went to the computer to explore the game in more detail. Before diving in, though, I had my son explain Carreira’s code:



Finally, we played the game a few times and looked at the different outcomes. The results really are amazing. Hopefully we’ll do a follow up project tomorrow:

Working through a Bayes’ Theorem problem thanks to help from Julia Anker

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:



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!

Revisiting Futility Closet’s “Paradox of the 2nd Ace” with my younger son

This week my younger son told me he wanted to learn a bit more about statistics. By lucky coincidence I happened to stumble on one of our old projects while trying to answer a question on twitter:

https://mikesmathpage.wordpress.com/2018/12/01/introducing-the-boys-to-futility-closets-paradox-of-the-second-ace/

This project was about a fun probability problem I learned in this tweet from Jon Cook:



I stared by introducing the problem to my son and asked what he thought the answer was going to be:



Then we started in on the calculations. Finding the probability that someone having “at least one ace” had more than one was a little challenging, but we found the right approach after a few tries:



Next up was calculating the probability that someone who had the ace of spades would have a second ace. After the work we did in the last part, this calculation was easier:



Finally, we went to the Futility Closet page to see the numbers and then I asked my son why he thought the surprising result was true.

Definitely a fun project and a neat probability surprise for kids to see.