Tag probability

Exploring Elchanan Mossel’s fantastic probability problem with kids

Saw a really great problem via a Lior Patcher tweet:

Here’s the problem:

You throw a dice until you get 6. What is the expected number of throws (including the throw giving 6) conditioned on the event that
all throws gave even numbers.

Here are direct links to Kalai’s two blog posts on the problem:

Gil Kalai’s “TYI 30: Expected number of dice rolls

Gil Kalai’s follow up post: Elchanan Mossel’s Amazing Dice Paradox (Your Answers to TYI 30)

It is fun to click through to the first Kalai blog post linked above to cast your vote for the answer if you haven’t seen the problem before.

We actually started the project today by doing that:

Next we rolled some 6-sided dice to see how this game worked. I note seeing the video that a few of the rolls went off camera, sorry about that 🙂

At the end we discussed what we saw and why what we found was a little surprising.

The next part of the project was having the boys play the game off camera until they found 5 rolls meeting the criteria.

After this exercise the boys started to gain some confidence that the answer to the problem was 3/2.

Now I walked them through what I think is the easiest solution to understand. It comes from a comment on the first Gil Kalai’s blog post linked above:

ProbabilityComment

Listening to this discussion now, I wish I would have done a better job explaining this particular solution. Still, I hope the discussion is instructive.

Finally, we went to Mathematica to evaluation the sum from the last video and then to explore the problem via a short program I wrote.

At the end of this video the boys some up their thoughts on the problem.

I love this problem. It isn’t that often I run across a clever problem that is interesting for both professional mathematicians and kids. Those problems are absolute

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Playing with Jim Propp’s essay on Arthur Engel

Jim Propp’s August 2017 blog post is absolutely terrific:

Prof. Engel’s Marvelously Improbably Machines

Even though we are visiting my parents in Omaha, I couldn’t resist having the boys watch the video in the essay and then play with the challenge problem.

Here’s what they thought after watching the video – the nice thing is that they were able to understand the problem Propp was discussing [also, I shot these videos with my phone, so they probably don’t have quote the quality or stability of our usual math videos]:

Next I had them play the game that Propp explained in his video. The idea here was to make sure that they understood how the [amazing!] solution to the problem shown in the video:

Next we tried the challenge problem from the essay. I almost didn’t do this part of the project, but I’m glad I did. It turned out that there were a few ideas in the Propp’s video that the boys thought they understood but there was a bit more explanation required. Once they got past those small stumbling blocks, they were able to solve the problem.

I’m really excited to dive a little deeper into the method of solving probability problems that Propp explains in his essay. What makes me the most excited is that the method came from someone thinking about how to explain probability to kids.

The last video shows that understanding Engel’s method does take a little time. Once kids get the general idea, though, I think they’ll find that applying to a wide variety of problems is pretty easy. It is amazing how such a simple method can made fairly complex probability problems accessible to kids.

Talking a bit more about my son’s probability problem

Yesterday we did a fun project on a probability problem / game my son was working on. The game involves rolling three 10-sided dice and adding up the numbers. Repeat the process until you’ve seen one of the sums 40 times. Yesterday 15 was the winner:

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Here’s yesterday’s project:

A probability and stats problem with dice my younger son was working on today

I wrote a short program on Mathematica to play my son’s game 1,000,000 times. I was interested to see how each of the boys would interpret the results.

Here’s what my older son had to say:

Here’s what my younger son had to say:

It is interesting to hear what kids have to say about the various probabilities and distributions. The results of the 1,000,000 simulations are probably pretty surprising. This problem that my son made up is actually a pretty fun problem to explore with kids.

A probability and stats problem with dice my younger son was working on today

When I got up this morning my younger son was playing some sort of dice game in the kitchen. An hour later he was still rolling dice so I finally asked him what he was doing:

It turns out what he was actually trying to was find the first sum that would appear 40 times, but I only understood that later.

This seemed like an easy activity to turn into a project, so we got started by having him explain what he was doing:

Next we turned to Mathematica to play around a little bit with the problem. I had to explain some terms first (and sorry I had part of the screen out of view for a bit). After explaining those terms we looked at the distribution of the sums:

Finally we wrapped up by taking a very deep dive into the distribution of the sum of three 10 sided dice. The kids were able to understand the probability of getting a 3 or a 30, and then we talked about a few of the other probabilities that Mathematica was showing us.

Later in the morning my son finished his game. 15 was the first roll to appear 40 times.

It was really fun to base a project on a math problem that my son came up with on his own.

The “Dungeons and Dragons” problem

My kids have suddenly been drawn into D&D. They are having a ton of fun with it and I thought that there was at least one fun little math problem we could talk through that related to the game.

At the start of the game you create a character with various properties. To determine the value of some (maybe all) of those properties you roll four 6-sided dice and add up the three highest numbers.

The question we looked at today was what is the expected value of that sum?

First we introduced the problem and come up with a few ideas about what the answer might be.

Next we did 10 trials to see what average we’d find:

Now we had a longish talk about how you might solve the problem. The boys jumped to a computer simulation pretty quickly. After talking about how that simulation would work we talked about how to solve a similar problem with two dice.

Finally, we did go the computer to see what the answer would be. Talking about how to write the program was pretty fun.

Nice project – I might revisit this one to talk through the geometry of the solution of the 2 and 3 dice problem and see if the boys can figure out how to generalize to the 4 dice case.

The coupon collection problem with kids

Yesterday my younger son was playing a dice game (explained in the first video) that reminded me a bit of the coupon collection problem. I thought it would be fun to try out that problem with the boys this morning. We were a little low energy, but I think it was still a good project. I’ll have to figure out how to revisit it to make sure the points stuck.

Here’s the introduction, including the game my son was playing:

Next we worked through one case of the problem – rolling dice trying to collect 6 “coupons”. My older son thought it would take 15 rolls and my younger son thought it was take 20.

Now I tried to help the kids dive into the math. We ended up going down a path that was much more complicated than I intended. I’m not sure why I made the choice that I did here, but . . . it happens sometimes 🙂

So, at the end of the last video we were caught in a seemingly complicated infinite series. I tried to explain why the expression we had on the board had to be equal to one. Then I tried to explain why the expected number of rolls had to be greater than one. The explanation here is a disaster, though.

Now that things had gone totally off the rails, I tried to pull it back. Luckily things did go better, and it was easier for the boys to see the expected number of rolls when there were fewer open slots.

Finally I wanted to show the kids how the ideas we talked about here would apply to a more difficult problem – say 100 coupons. We got off on the wrong foot here, but we eventually saw how the ideas we’d talked about previously applied.

Despite the low energy and going doing a path that was a bit too complicated, I think this is a fun problem for kids to study. It looks very difficult initially, but through a bit of calculation (and maybe a bit of hand waving) we can break it down into some smaller problems that we are able to solve. Putting the solutions of those smaller problems together, we can show that the solution to the original coupon collection problem isn’t too hard to understand.

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Connecting yesterday’s probability project with a few old 3d geometry projects

In yesterday’s project we were studying a fun probability question posed by Alexander Bogomolny:

That project is here:

Working through an Alexander Bogomolny probability problem with kids

While writing up the project, I noticed that I had misunderstood one of the
geometry ideas that my older son had mentioned. That was a shame because his idea was actually much better than the one I heard, and it connected to several projects that we’ve done in the past:

Paula

 

Learning 3d geometry with Paula Beardell Krieg’s Pyrmaids

Revisiting an old James Tanton / James Key Pyramid project

Overnight I printed the pieces we needed to explore my son’s approach to solving the problem and we revisited the problem again this morning. You’ll need to go to yesterday’s project to see what leads up to the point where we start, but the short story is that we are trying to find the volume of one piece of a shape that looks like a cube with a hole in it (I briefly show the two relevant shapes at the end of the video below):

Next we used my son’s division of the shape to find the volume. The calculation is easier (and more natural geometrically, I think) than what we did yesterday.

It is always really fun to have prior projects connect with a current one. It is also pretty amazing to find yet another project where these little pyramids show up!

Working through an Alexander Bogomolny probability problem with kids

Earlier in the week I saw Alexander Bogomolny post a neat probability problem:

There are many ways to solve this problem, but when I saw the 3d shapes associated with it I thought it would make for a fun geometry problem with the boys. So, I printed the shapes overnight and we used them to work through the problem this morning.

Here’s the introduction to the problem. This step was important to make sure that the kids understood what the problem was asking. Although the problem is accessible to kids (I think) once they see the shapes, the language of the problem is harder for them to understand. But, with a bit of guidance that difficulty can be overcome:

With the introduction out of the way we dove into thinking about the shape. Before showing the two 3d prints, I asked them what they thought the shape would look like. It was challenging for them to describe (not surprisingly).

They had some interesting comments when they saw the shape, including that the shape reminded them of a version of a 4d cube!

Next we took a little time off camera to build the two shapes out of our Zometool set. Building the shapes was an interesting challenge for the kids since it wasn’t obvious to them what the diagonal line segments should be. With a little trial and error they found that the diagonal line segments were yellow struts.

Here’s their description of the build and what they learned. After building the shapes they decided that calculating the volume of the compliment would likely be easier.

Sorry that this video is a little fuzzy.

Having decided to look at the compliment to find the volume, we took a look at one of the pieces of the compliment on Mathematica to be sure that we understood the shape. They were able to see pretty quickly that the shape had some interesting structure. We used that structure in the next video to finish off the problem:

Finally, we worked through the calculation to find that the volume of the compliment was 7/27 units. Thus, the volume of the original shape is 20 / 27.

As I watched the videos again this morning I realized that my older son mentioned a second way to find the volume of the compliment and I misunderstood what he was saying. We’ll revisit this project tomorrow to find the volume the way he suggested.

I really enjoyed this project. It is fun to take challenging problems and find ways to make them accessible to kids. Also, geometric probability is an incredibly fun topic all by itself!

James Tanton’s counting problem part 2

Yesterday we looked at a really neat problem James Tanton posted last week:

That project is here:

Working through a challenging counting problem from James Tanton

Our first look at the question involved some dice rolling and a computer simulation. Today we are going to look at an exact solution to the problem. That solution involves studying all of the different things that can happen when you roll 5 dice. It turns out that there are 7 different patterns that can happen, and these patterns related to the ways you can write 5 as the sum of positive integers.

(1) 5 different numbers, which I’ll represent as 1 + 1 + 1 + 1 + 1

(2) 3 different and 2 the same -> 1 + 1 + 1 + 2

(3) 2 different and 3 the same -> 1 + 1 + 3

(4) 1 different and 4 the same -> 1 + 4

(5) 1 different and 2 pairs -> 1 + 2 + 2

(6) 1 pair and 1 triple -> 2 + 3

(7) All numbers the same -> 5

For today’s project we’ll count the number of ways that each of these 7 patterns can occur. We know that the total number of arrangements is 7,776, so that’s going to help us make sure we have counted correctly.

Here’s the introduction to the problem and to the approach we are going to take today:

Now we began to count some of the arrangements. In this video we count the number of dice rolls in (1), (4), and (7) above:

Now we moved on to some of the more challenging arrangements. Here we looked at (6) and (2) above:

Now we looked at case (5). This case proved challenging because dealing with the 2 pairs caused a little confusion between over counting and under counting. But, after looking at the cases carefully we did manage to get to the answer.

At this point we had only one case left -> (3) from above. But, the counting practice that we’d had up to this point helped this case go pretty quickly.

Finally, we added up our numbers and checked that we’d found all 7,776 cases. We did!

The one thing left to do was to count the different numbers that we saw in each case and find the average. I’d done that ahead of time just to save a bit of time in the movie. Our final answer was (27,906) / (7,776) or about 3.588. The exact answer was (happily!) very close to the two estimates that we had found in our simulations yesterday.

I love Tanton’s problem. It is a great estimation problem as well as a great counting problem. We might do one more project tonight on yet a different way to solve the problem using Markov chains:

Looks like a fun idea – I’ll be thinking about how to talk through this approach with the kids during the day today.

A neat expected value problem from Expii

[sorry for the quick write up – I got asked to help out with my son’s archery class today, so I just decided to publish this one as it was when I get asked to help . . . ]

I saw a neat expected value problem from Expii yesterday. In case you’ve not see their site, here’s the link to their main site:

Expii’s front page

and here’s a direct link to the problem:

A neat expected value problem from Expii

The problem goes like this:

“You are planting some trees as environmental action for Earth Day. At each of 200 spots around a circle, you place a seed. Each seed will sprout into a small tree with probability 1/2. Sadly, some of these small trees will die. In particular, a small tree dies if it has another small tree as its neighbor, because they will be fighting for sunlight.

What is the expected value of the number of trees that are still alive at the end of the year?”

I thought this would be a great problem to discuss with the boys. We just got back from a vacation in San Diego and my younger son was still on west coast time, though, so I just talked through this one with my older son.

First I introduced the problem and we double checked that he understood it:

Next we discussed some simple cases to see if we could get our arms around the problem:

Now we moved on to the general case. My son understood some of the main ideas about the problem, but made a small mistake at the end that led to a very small expected value.

Finally, we wrapped up by looking at the error at the end of the last video and trying to calculate the expected value slightly more carefully: