Sharing problem #21 from Mosteller’s Fifty Challenging Problems in Probability with my kids

I found a neat problem in Mosteller’s Fifty Challenging Problems in Probability to share with the boys this morning:

In terms of “what kids learning math look like” this is one of the most interesting projects we’ve done. The problem turned out to be basically accessible to the boys, but pretty close to the end of what they could solve. Eventually they got there, though. I feel like this post could be useful for both the math and for seeing how kids think about math as they are learning it.

I started with a quick introduction to the problem and asked for their initial thoughts:

The first thing the boys wanted to do is look at the replacement and no replacement strategies with the urn with three blocks:

Next they looked at the two strategies in the urn with 201 blocks. We pretended the chances of each block were 50/50 to simplify the math a little:

Now things started going in a direction that was a bit different from what I was expecting. I thought we were just about to wrap up, but it turned out that the right way forward wasn’t yet clear in their minds. But even though things aren’t yet clear to them, you’ll see that they are making progress towards the answer:

Next we were able to combine the two decisions trees and start looking at how often our decision rule would lead us to picking the correct urn. Towards the end of this video suddenly the ideas in the problem become clear to the boys:

Here we finish the calculation of guessing right with our strategy – with the problem now making sense to them the calculation goes pretty quickly:

Sharing Zachary Binney’s Twitter thread on test sensitivity and specificity with kids

I saw a neat twitter thread from Zachary Binney last week:

The ideas in Binney’s thread are really important if you want to understand testing, so I thought I’d share them with the boys this morning. We started by looking at the thread and then going to Wikipedia to get a few definitions:

Now we went through a few specific examples. For all three we assumed the test was 95% accurate. In our first example we assumed that 5% of the population would have a disease. What is your chance of having the disease if you test positive?

Next we looked at what would happen if only 1% of the population has the disease (sorry the camera wasn’t showing the bottom of the white board here 😦 ):

Finally we looked at what would happen if 30% of the population had the disease:

The problem we are looking at here is a pretty famous one in probability and statistics. Binney’s twitter thread made for a great opportunity to show how the ideas aren’t just theory or problem set problems, too.

Sharing an interesting (and famous) population sampling problem with kids

I saw a great thread on twitter last week – actually in the reverse order in which the tweets appeared. First I saw from Vincent Pantaloni:

Which led me to this amazing tweet from Andrew Webb:

I thought it would be fun to do a project on this idea with the boys. Unlike a few (or maybe most) of our introductory statistics exercises, the program here was likely going to be too hard for the kids to write themselves, so I just wrote it myself and the boys played with it at the end.

To start I had them look at Pantaloni’s tweet:

Next we looked at Webb’s tweet – this one requires a bit more explanation, but the boys were able to understand what Webb’s animation was showing:

Now I spent 5 min explaining how the program I wrote worked. Since my simulation was quite a bit more simplified than the prior two (and also didn’t have any animation), I wanted to be sure they understood what I was doing before we dove in:

For the first run of my simulation, we looked at a 5000 trials of a pond with 1,500 fish and sampling from 4% of the pond From the conversation here you can hear that the boys are gaining a pretty good understanding of the process and are also able to make sense of the distribution of outcomes:

Finally, we looked at 5000 trials of a pond with 750 fish and sampling from 16% of the pond. Again the boys did a nice job explaining the results.

At the end we talked about why this sort of sampling problem can be really difficult.

Exploring some probability ideas from poker played with 2 decks of cards with kids

On New Year’s eve my older son was playing poker with his friends using 2 decks of cards – see if you can spot the oddity in the picture 🙂

For today’s project we decided to explore some of the probability ideas around playing poker with 2 decks of cards. First we just looked at the possible hands and talked about some potential questions to ask:

For an introductory problem, we looked at the number of ways of getting a Royal Flush and then all types of flushes with a 5 card hand dealt from a single deck of cards:

Now we looked at how regular flushes could happen when dealt from a two decks of cards shuffled together. We also had a good discussion about whether or not it was more likely or less likely to get a flush in the 2 deck situation:

Finally, we went back to Mathematica to take a look at the numbers for a general flush with one and two decks. Here we are lumping all kings of flushes together – regular ones, straight flushes, and Royal flushes are all the same.

Compute the chances of various hands with 2-deck poker is a pretty fun math exercise for kids.

Two probability problems that seem similar but have different answers

Earlier in the week we looked at the game Ox Blocks which uses a 6-sided die with 2 sides each having O’s, X’s, and a blank. The game is a really fun version of tic tac toe:

Playing with Ox Blocks thanks to the Mathematical Objects podcast

Playing this game reminded me of an old project we’d done on a fun probability problem from Elchanan Mossel:

Exploring Elchanan Mossel’s fantastic probability problem.

For today’s project we looked at two problems inspired by these two projects. The problems seem pretty similar:

(1) If you have a fair 6-sided die with sides marked 2, 2, 4, 4, 6, and 6, how many rolls on average will it take for you to roll a 6.

(2) If you have a fair 6-sided die with sides marked 1, 2, 3, 4, 5, and 6, how many rolls on average will it take to roll a 6 if any sequence of rolls containing an odd number prior to seeing a 6 doesn’t count. So, 2, 4, 4, 6 would count, for example, and 2, 4, 5, 6 would not count.

I started the project today looking at the first problem, which is inspired by the Ox Blocks project:

Now we moved to the 2nd problem. To introduce the problem I had the boys play the game a few times and we found that lots of sequences of rolls were thrown out:

To help the boys understand this second game a bit more I moved to a slightly different question -> for valid sequences of rolls in the 2nd game, how often do you see a 6 on the first roll.

This question was slightly difficult for the kids to understand, but we made pretty good progress:

Finally, we went to the computer to run a simulation for the 2nd game. This video runs a little long as I asked my younger son to explain the program. But once we get through the explanation we see that their guesses for the expected number of rolls and also the percentage of 6’s on the first roll were roughly right!

Writing a program to see how long it takes to get HHHH when flipping a coin

Yesterday we did a fun project exploring how long it takes, on average, to create certain words like COVFEFE or ABRACADABRA when selecting letters at random. We also simplified the problem a bit by looking at sequences of H’s and T’s for coin flips. That project is here:

Talking Markov chains and Martingales with kids

Today’s project was writing a computer program to simulate flipping a coin until we saw HHHH. In yesterday’s project we found that it would take 30 coin flips on average. We started today’s project by talking about how to write a program to do this simulation. Following this discussion the boys wrote their program off camera:

When the boys finished their program we talked through it and looked at the shape of the distribution of the number of flips it took to get to HHHH. They were pretty surprised by this shape:

To wrap up the project we spent 5 min talking about how the program would need to change to look at a general sequence of 4 flips – HTHT, for example. We didn’t actually make the changes, though, as we’d already spent enough time working through the ideas this morning:

This was a fun statistics / programming project that has a pretty surprising result. We’ll definitely have to follow up with the program for a generic sequence of flips soon!

Exploring Markov chains and Martingales with kids

Earlier this week I saw a neat tweet from Greg Egan:

It reminded me of an old project we did back in 2017 using Markov chains and Martingales:

The Most Interesting Piece of Math I Learned in 2017 -> The COVFEFE Problem

The Martingale take on the COVFEFE problem (and the amazing ABRACADABRA problem) came from this paper:

Martingale’s and the ABRACADABRA problem by Di Ai

This week I was using some of the Markov chain ideas for a few fun projects with my older son who is studying linear algebra. Today I though it would be fun to revisit the COVFEFE problem and then look at some coin flipping examples inspired by Greg’s tweet.

We started with a brief discussion of the COVFEFE problem and then switched to coin flipping at the end. It took me a bit to get my brain going on this project – sorry for a few obvious mistakes at the beginning of the discussion . . . .

Next we went to the computer to look at the approach to the word typing / coin flipping problems via Markov chains. Instead of the COVFEFE problem, we are looking at the expected number of coin flips required to see the sequence HTHT. The ideas here and the code are things I learned from Nassim Taleb – see the references in the project linked above:

Next we returned to the whiteboard to talk about the Martingale approach to the problem. The ideas here are things that I learned from Christopher Long and also from the paper linked above.

Here we take a quick look at the COVFEFE problem, the ABRACADABRA problem, and see why the HTHT problem takes on average 20 flips.

Finally, we computed the expected number of flips required to see each of the 16 different combinations of 4 coin flip sequences -> HHHH, HHHT, HHTH, and etc. The calculation for all 16 cases took a little longer than I usually want one of these videos to run, but I wanted to do all of the cases to help the boys understand the ideas that go into the calculations:

A detailed discussion of the concepts of both Markov Chains and Martingales are above what anyone could reasonably expect a 10th grader and 8th grader to understand. But the ideas are so neat that I thought showing these fun examples would make a great project for kids.

Sharing Problem 10 from Mosteller’s 50 Challenging Problems in Probability with my younger son

Today my older son is at an event, so the project was just with my younger son. The project is a version of a famous problem attributed to Daniel Ellsberg (Mosteller’s book also credits Ellsberg). Here’s the Wikipedia page on the more famous problem:

I got start with today’s project by reading the first problem and having my son share his initial thoughts about how to approach solving it:

With some thoughts down on the white board, now we turned to solving the problem. I loved his thought process here:

Next we moved on to problem attributed to Ellsberg – this gives problem is a fun twist on the first problem since now you do not know how many balls of each color are in the bag:

With my son having written down a few ideas about the new problem in the last video, he now gave his solution to the 2nd problem. His thinking here is also really great. It is fun to see young kids talk through a difficult problem like this one:

What are the chances of a class with 24 students having 3 pairs of students sharing a birthday?

Yesterday we looked at the famous Birthday problem – how many people do you need to have in a room to have a 50/50 chance of two people having the same birthday? That project is here:

Diving into the Birthday problem with kids

Today we continued the project (with just my older son as my younger son was hiking) and studied the problem that originally motivated this project -> If you have 24 students in a class, what is the chance that exactly 3 pairs of students will share a birthday? This is the surprisingly fun situation in my son’s English class.

We will – as I think it standard for the introductory version of this problem – be making the assumption that all birthdays are equally likely. If you want to see a really neat discussion – though not really a math for kids paper! – see the paper in this tweet:

So, to start the project today we first reviewed the main ideas from yesterday:

Next we took a step towards solving the problem by looking at the chance of having exactly 2 pairs. Once piece of the counting here is tricky, so we used the computer to help see what the problem was.

Now we tackled the “exactly 3 pairs problem”:

Finally, I had my son make up a problem to solve – he decided to find the chance of all 24 students pairing up. This problem wasn’t too hard given the prior work. It was also a fun challenge to try to estimate the chance of this happening.

Diving into the Birthday problem with kids

Earlier in the week I learned that my older son’s high school English class has 24 students and 3 pairs of students who share the same birthday. None are twins, so no tricks or anything like that, just a fun fact for this particular class.

I thought it would be fun to figure out how rare something like this would be – assuming, of course, that all of the birthdays are randomly distributed amount the 366 possible birthdays (366 because many of the kids were born in 2004).

It turns out the chance of having exactly three pairs of kids with the same birthday (and no other shared birthdays) in a class of 24 kids is roughly 2.3%, or if you prefer the exact answer:


Instead of continuing with Mosteller’s book this weekend, I thought it would be fun to dive into the birthday problem. I started today with the standard problem – how many people do you need in a room for a 50% chance of two people sharing the same birthday. This is not an easy problem and the answer is not intuitive.

Here’s how we got started – not surprisingly, down a path that wasn’t quite right:

After coming up with a formula in the last video, we went to Mathematica to see what it said. Here we discovered that the formula was giving answers that were not correct:

Now we returned to the whiteboard and the boys found a new formula – this one calculated the chance of having exactly 1 pair with the same birthday. I was happy that they were able to derive this formula and even happier for the chance to show them it didn’t agree with our computer modelling!

Now we went back to the computer to see the surprise that our new – and much closer to correct – formula actually didn’t agree with the modelling. What was wrong?

Finally, having figured out why the two approaches didn’t match, the boys were able to find the correct formula to solve the problem. Tomorrow we’ll dive into the more complicated problem of finding the probability of 3 pairs: