Tag probability

Sharing an advanced expected value problem from Nassim Taleb with kids

Earlier in the week I saw this interesting problem posted by Nassim Taleb:

Solving this problem requires calculus, and trig to even begin to understand how to approach it, but it still seemed like one that would be interesting to talk through with kids. Especially since a Monte Carlo-like approach is going to lead you down a surprising path.

So, I presented this problem to the boys this morning. It took a few minutes for them to get their arms around the problem, but they were able to understand the main ideas behind the question. That made me happy.

Here’s the introduction to the problem:

Next I asked the boys what they thought the answer to this question would be. It was fascinating to hear their reasoning. Both kids had the same guess -> the expected average distance was 1.

Now we went to the computer to see what the average was when we did a few trials. We started by doing 100 trials to estimate the average and then moved up to 10,000 trials.

Next we went to 1 million trials and found a few big surprises including this amazing average:

Screen Shot 2018-04-08 at 9.20.06 AM.png

We wrapped up by discussing how you might get an infinite expected value by looking at the values of Tan(89), Tan(89.9), Tan(89.99), and so on. It was interesting for them to see how individual trials could have large weights, even with large numbers of trials.

Definitely a fun project to show kids, and a nice (though advanced) statistics lessonm too -> What happens when the mean you are looking for is infinite?

Advertisements

15 (+1 bonus) Math ideas for a 6th grade math camp

Saw an interesting tweet last week and I’ve been thinking about pretty much constantly for the last few days:

I had a few thoughts initially – which I’ll repeat in this post – but I’ve had a bunch of others since. Below I’ll share 10 ideas that require very few materials – say scissors, paper, and maybe snap cubes – and then 5 more that require a but more – things like a computer or a Zometool set.

The first 4 are the ones I shared in response to the original tweet:

(1) Fawn Nguyen’s take on the picture frame problem

This is one of the most absolutely brilliant math projects for kids that I’ve ever seen:

When I got them to beg

Here’s how I went through it with my younger son a few years ago:

(2) James Tanton’s Mobius strip cutting exerciese

This is a really fun take on this famous scissors and paper cutting exercise:

You will honestly not believe what you are seeing when you go through Tanton’s version:

Here’s the link to our project:

James Tanton’s incredible mobius strop cutting project

(3) Martin Gardner’s hexapawn “machine learning” exercise

Screen Shot 2017-09-03 at 10.07.08 AM

For this exercise the students will play a simple game called “hexapawn” and a machine consisting of beads in boxes will “learn” to beat them. It is a super fun game and somewhat amazing that an introductory machine learning exercise could have been designed so long ago!

Intro “machine learning” for kids via Martin Gardner’s article on hexapawn

(4) Katie Steckles’ “Fold and Cut” video

This video is a must see and it was a big hit with elementary school kids when I used it for “Family Math” night:

Here are our projects – all you need is scissors and paper.

Our One Cut Project

Fold and cut project #2

Fold and cut part 3

(5) Along the same lines – Joel David Hamkins’s version of “Fold and Punch”

I found this activity in one of the old “Family Math” night boxes:

Joel David Hamkins saw my tweet and created an incredible activity for kids.ย  Here’s a link to that project on his blog:

Joel David Hamkins’s fold, punch and cut for symmetry!

(6) Kelsey Houston-Edwards’s “5 Unusual Proofs” video

Just one of many amazing math outreach videos that Kelsey Houston-Edwards put together during her time at PBS Infinite Series:

Here is how I used the project with my kids:

Kelsey Houston-Edwards’s “Proof” video is incredible

(7) Sharing the surreal numbers with kids via Jim Propp’s checker stacks game

Screen Shot 2018-03-24 at 7.52.55 PM

Jim Propp published a terrific essay on the surreal numbers in 2015:

Jim Propp’s “Life of Games”

In the essay he uses the game “checker stacks” to help explain / illustrate the surreal numbers. That essay got me thinking about how to share the surreal numbers with kids. We explored the surreal numbers in 4 different projects and I used the game for an hour long activity with 4th and 5th graders at Family Math night at my son’s elementary school.

This project takes a little bit of prep work just to make sure you understand the game, but it is all worth it when you see the kids arguing about checker stacks with value “infinity” and “infinity plus 1” ๐Ÿ™‚

Here is a summary blog post linking to all of our surreal number projects:

Sharing the Surreal Numbers with kids

(8) Larry Guth’s “No Rectangle” problem

I learned about this problem when I attended a public lecture Larry Guth gave at MIT.ย  Here’s my initial introduction of the problem to my kids:

I’ve used this project with a large group of kids a few times (once with 2nd and 3rd graders and it caused us to run 10 min long because they wouldn’t stop arguing about the problem!). It is really fun to watch them learn about the problem on a 3×3 grid and then see if they can prove the result. Then you move to a 4×4 grid, and then a 5×5 and, well, that’s probably enough for 80 min ๐Ÿ™‚

Larry Guth’s “No Rectangles” problem

(9) The “Monty Hall Problem”

This is a famous problem, that equally famously generates incredibly strong opinions from anyone thinking about it. These days I only discuss the problem in larger group settings to try to avoid arguments.

Here’s the problem:

There are prizes behind each of 3 doors. 1 door hides a good prize and 2 of the doors hide consolation prizes. You select a door at random. After that selection one of the doors that you didn’t select will be opened to reveal a consolation prize. At that point you will be given the opportunity to switch your initial selection to the door that was not opened. The question isย  -> does switching increase, decrease, or leave your chance of winning unchanged?

One fun idea I tried with the boys was exploring the problem using clear glasses to “hide” the prizes, so that they could see the difference between the switching strategy and the non-switching strategy:

Here’s our full project:

Exploring the Monty Hall problem with kids

(10) Using the educational material from Moon Duchin’s math and gerrymandering conference with kids

Moon Duchin has spent the last few years working to educate large groups of people – mathematicians, politicians, lawyers, and more – about math and gerrymandering.ย  . Some of the ideas in the educational materials the math and gerrymandering group has created are accessible to 6th graders.

Screen Shot 2018-01-14 at 9.08.06 AM

Here’s our project using these math and gerrymandering educational materials:

Sharing some ideas about math and gerrymandering with kids

(11) This is a computer activity -> Intro machine learning with Google’s Tensorflow playground.

This might be a nice companion project to go along with the Martin Gardner project above. This is how I introduced the boys to the Tensorflow Playground site (other important ideas came ahead of this video, so it doesn’t stand alone):

Our complete project is here:

Sharing basic machine learning ideas with kids

(12) Computer math and the Chaos game

The 90 seconds starting at 2:00 is one of my all time favorite moments sharing math with my kids:

The whole project is here, but the essence of it is in the above video:

Computer math and the chaos game

(13) Another computer project -> Finding e by throwing darts at a chess board

This is a neat introductory probability project for kids. I learned about it from this tweet:

You don’t need a computer to do this project, but you do need a way to pick 64 random numbers. Having a little computer help will make it easier to repeat the project a few times (or have more than one group work with different numbers).

Here’s how I introduced the project to my kids:

Here’s the full project:

Finding e by throwing darts

(14) Looking at shapes you can make with bubbles

For this project you need bubble solution and some way to make wire frames. We’ve had a lot of success making the frames from our Zometool set, but if you click through the bubble projects we’ve done, you’ll see some wire frames with actual wires.

Here’s an example of how one of these bubble projects goes:

And here’s a listing of a bunch of bubble projects we’ve done:

Our bubble projects

(15) Our project inspired by Ann-Marie Ison’s math art:

This tweet from Ann-Marie Ison caught my eye:

Then Martin Holtham created a fantastic Desmos activity to help explore the ideas:

It is fun to just play with, but if you want to see how I approached the ideas with my kids, here are our projects:

Using Ann-Marie Ison’s incredible math art with kids

Extending our project with Ann-Marie Ison’s art

(16) Bonus project!!A dodecahedron folding into a cube

This is a an incredible idea from 3d geometry.

We studied it using our Zometool set – that’s not the only way to go, but it might be the easiest:

dodecahedron fold

Here’s the full project:

Can you believe that a dodecahedron folds into a cube?

A terrific probability problem for kids shared by Alexander Bogomolny

Saw this tweet from Alexander Bogomolny yesterday and knew immediately what today’s project was going to be ๐Ÿ™‚

The problem is, I think, accessible to kids without much need for additional explanation, so I just dove right in this morning to see how things would go.

My first question to them was to come up with a few thoughts about the problem and some possible strategies that you might need to solve it. They had some good intuition:

Next we attempted to use some of the ideas from the last video to begin to study the problem. Pretty quickly they saw that the initial strategy they chose got complicated, and a more direct approach wasn’t actually all that complicated:

I intended to have them solve the 4x4x4 problem with one of our Rubik’s cubes as a prop, but we could only find our 5x5x5 cube. So, we skipped the 4x4x4 case, solved the 5x5x5 case and then jumped to the NxNxN case:

Finally, I wanted the boys to see the “slick” solution to this problem – which is really cool. You’ll hear my younger son say “that’s neat” if you listen carefully ๐Ÿ™‚

Definitely a fun problem – would be really neat to share this one with a room foll of kids to see all of the different strategies they might try.

Sharing a great Alexander Bogomolny probability problem with kids

[note: I’m trying up this post at my son’s karate class. It is loud and unfortunately I forgot my headphones. I’m left having to describe the videos without being able to listen to them . . . . ]

I saw a really great problem today from Alexander Bogomolny:

By coincidence I heard the recent Ben Ben Blue podcast yesterday which had a brief mention / lament that it was hard to share mistakes in videos.

This problem is probably a good challenge problem for my older son and definitely above the level of my younger son. But listening to both of them try to work through the problem was really interesting.

I started with my older son – he initially approached the problem by comparing the individual probabilities:

After his initial work, I talked with him about comparing the probabilities of the complete events described in the problem. Initially there was a little confusion on his part, but eventually he understood the idea:

Next up was my younger son – not surprisingly, he had a hard time getting started with the problem. His initial approach was similar to what my older son had done – he looked at the one head and two heads events separately to see which one was more likely for each coin:

As I did with my older son, I asked him to look at the two events as a single event and see which one was more likely when each coin went first:

So, a nice project and an opportunity to see a few mistakes and as well as how kids approach a challenging probability problem.

A nice probability coincidence today

This morning my older son and I talked through the following problem from the 2003 AMC 10b:

Screen Shot 2018-02-15 at 7.11.31 PM.png

It turned out that an arithmetic problem is what led to his confusion on this problem, but discussing this problem was a nice opportunity to talk about discrete probability. At the end of our conversation I told him to always remember the underlying idea in discrete probability is simple -> count the cases that work and then count the total cases. It may not always be so easy to do, but it really is all that you have to do to solve the problem.

After he went off to school I this problem posted on Twitter:

This is a terrific problem, and it is really tempting to try to break the problem into pieces and essentially to try to solve it with recursion.

But remember the simple little idea I told my son -> count the cases that work and count the total cases. You’ll find a delightful solution to the problem!

Talking through two problems from the 2005 AMC 10

I really enjoy using old AMC problems to talk about math with the boys.

These two problems gave my older son some trouble today:

Tonight I had a chance to talk through these problems with them.

Here’s the probability problem:

Here’s the GCD problem:

Sharing the ABRACADABRA problem with kids

Yesterday we did a fun project on Markov chains and sharing the “COVFEFE” problem with kids:

Sharing Markov chains and the “covfefe” problem with kids

For me math behind this problem was the most interesting math I learned in 2017:

The most interesting piece of math I learned in 2017 -> the “covfefe” problem

Today we moved on to a really neat surprise, and what makes the math behind this problem incredibly fun -> the “ABRACADABRA” problem.

First, we reviewed the ideas from yesterday:

After that review, we though through a few of the states and the transition probabilities in the new word. The transition probabilities are subtly different than in the “COVFEFE” problem:

Now we went to Mathematica to code in the ideas we discussed in part 2. We did about half of the coding on camera and did the other half off camera:

Finally, having finished the code we discussed what results we expected. I don’t see how anyone could get the right intuition here seeing the problem for the first time, so what do you expect here is almost an unfair question. Still, the boys had some nice ideas and then we checked out the results:

There are other approaches to these problems – the approach via Martingales, for example:

What that approach is also interesting (and incredible – you can solve the stopping time in your head!) I think the Markov chain approach is a bit more accessible to kidsd. Well . . . maybe because the math is buried in the background.

Anyway – super fun project, and an great piece of math to share with kids.

Sharing Markov chains and the “COVFEFE” problem with kids

In 2017, the most interesting piece of math I learned can via Christopher Long and Nassim Taleb and related to the “COVFEFE” problem:

Screen Shot 2017-12-11 at 5.05.10 AM

I wrote about the problem here:

The most interesting piece of math I learned in 2017 -> the “covfefe” problem

It turned out that we’ve looked at Markov chains before thanks to this great video from Kelsey Houston-Edwards:

Sharing Kelsey Houston-Edward’s Markov chain video with kids

I’d forgotten about that project, but when I mentioned to my younger son that we’d be looking at Markov chains today he told me he already knew about them!

So, I started today by having the boys watch the PBS Infinite Series video again. Here’s what they thought:

Next I introduced the “COVFEFE” problem. I was really happy how quickly the boys were able to pick up on how Markov chains could be used to solve this problem.

Next we looked at Nassim Taleb’s Mathematica code – that code is so clear that the problem becomes instantly accessible to kids, which is pretty amazing.

Finally, since things were going so well this morning, I introduced the word that we’ll study tomorrow -> ABRACADABRA. The kids were able to see why the transitions in this word were different. I’m excited to see how they think through the “ABRACADABRA” problem tomorrow!

The math behind this problem really was the most interesting math that I learned in 2017. It is really important math, too, and I’m excited that the Mathematica code makes some of the ideas accessible to kids. This was a fun one!

The most interesting piece of math I learned in 2017 -> the COVFEFE problem

This question from a math exam in India was flying around math twitter last week:

Screen Shot 2017-12-11 at 5.05.10 AM.png

I mistakenly thought the question was just sort of a fun joke and not all that interesting, but then I saw a series of tweets beginning with:

and ending with:

That last tweet was definitely a “wait . . . what??” moment for me.

Thinking about ABRACADABRA right off the bat was too hard, so I simplified the problem drastically too see what was going on. Suppose you are flipping a fair coin, what is the expected number of flips until you see the sequence H H? What about H T? These two problems are also, I think, great ways to introduce ideas about stopping time problems to kids. (the answers are 5 flips and 4 flips respectively).

Playing around with the easier problems showed me why the ABRACADABRA problem could have a longer stopping time than I would have guessed, but I couldn’t solve the problem exactly. Then I found this paper (written as part of an undergraduate research program at the University of Chicago!) which gave a wonderful explanation of the ABRACADABRA problem and (almost incredibly) a way to think about the problem that allows you to solve it in your head!

Screen Shot 2017-12-11 at 5.18.38 AM

Martingale’s and the ABRACADABRA problem by Di Ai

After going through that paper I was happy to have learned some new ideas about stopping time problems and more or less moved on. But then one more nice surprise came from the COVFEFE problem when Nassim Taleb shared his Markov chain solution:

I’d never played with Markov chains in Mathematica (or, basically anywhere really) so I thought it would be fun to use what I learned from Taleb’s code to explore the ABRACADABRA problem. Working through that code gave me a much better understanding of Long’s “which states lead to which states” comment above. It took me a bit of time to realize, for example, that the state ABRA can move to the state AB, for example.

Again, copying Taleb’s code, here’s the transition matrix:

Transition States

and the graph of the states plus the stopping time which matches 26^11 + 26^4 + 26:

Time to Stop.jpg

It was fun to learn that the original COVFEFE problem was part of a class of problems that are much more subtle than they might seem at first glance. Learning about the connections to martingales and learning how to implement Markov chains in Mathematica was a really nice surprise, too.

Revisiting non-transitive dice

We’ve done a few projects on non-transitive dice in the past:

Sharing Tim Gower’s non-transitive dice talk with kids

Non-Transitive Grime Dice

Today we’ve got some snow to shovel, so I was looking for a fairly light project this morning so we could get out the door to shovel. I grabbed our Grime dice off of the shelf and asked the kids to talk about them:

I asked the boys to pick two pairs of dice and test them to see which color would win. They worked independently and here’s how they explained what they found:

Finally, for a bit of a challenge, I had them work together to put the dice in a circular arrangement so that every color beat the one coming after it and lost to the one before it. This arrangement illustrates the seemingly odd non-transtive nature of these dice:

Although short, this was a fun exercise. These “Grime” dice are really fun for kids to play with!