I was re-reading How to Gamble if you Must by Dubins and Savage and though it would be fun to talk through the gambling problem in the beginning of the book with my younger son.
The book isn’t exactly light reading, but definitely interesting if you want to understand a bit of the math behind gambling
The problem is fairly straightforward to understand -> you start with $1,000 and you need to get to $10,000 by making bets. How should you bet if the game is unfavorable, fair, and favorable?
I started the project today by explaining the game to my son and asking how he thought you should bet in the various games:
Next we wrote a short program in Mathematica (off camera) and then played the game. Here’s a discussion of the program and a few times through the unfavorable game:
Now we played the fair game and looked to see if the strategy was the same or different than the unfavorable game:
Finally, we played the favorable game – again we looked for what might be different in the betting strategy for this game:
Via my friend Ed Adlerman, I saw some amazing temperature graphs made by Deke Arndt:
I thought these graphs could be used for a great introductory statistics talk for kids, so I decided to used them to talk about probability distributions today.
We started by looking a probability distributions in a relatively simple situation -> dice:
Next we moved to talking about a probability distribution in a more complicated situation -> a “Galton board”:
Now we moved on to discussing Arndt’s graphs. The conversation about these graphs went on for 11 minutes. As always it is fascinating to hear what kids see when they look at advanced math.
Here’s the first part of that 11 min discussion – here you’ll hear their initial observations and a bit about how to interpret the distributions on display here.
Here’s the second part of our conversation about the graphs. Here we talk about what all of these pictures are telling us about the temperature in Sydney during the last 60 years.
We wrap up by talking about whether or not they liked this presentation and different ways to present the data that would have been made the presentation harder to understand (I though this would be an easier question for kids than finding ways to make it better).
I think Arndt’s work here is amazing all by itself, but is also something that I think can be used really effectively to talk about probability and statistics with kids. Thanks to him for sharing this great work.
Saw a fun tweet from Bob Lochel last week:
For today’s Family Math problem I thought I’d talk through the math behind this situation with my younger son.
First I showed him what had happened and asked for a few ideas about how to approach finding the probability that all of the Matt’s would end up in group d. It has been a while since we’ve done a probability and counting project, but the ideas came back to him as we talked:
Next we talked about how to calculation the number of different ways to put the students into the the groups of 5 and 4. Then I asked my son to estimate what the number was without calculating. His estimate was off and we returned to why in part 4 of the project:
Now we calculated the number of ways to arrange the students with all 3 Matt’s landing in group D. Once we had that number we found the probability of the outcome that happened. I also asked my son if he thought the outcome happened by random chance, or was Bob possibly playing a little joke:
Finally, we revisited the calculation from part 2 of the project. It turned out that his estimate actually was nearly right – investigating where it went wrong was a good use of time. Once we had the exact value by hand, we also computed the exact probability for the original problem by hand.
Thanks to Bob Lochel for sharing this fun outcome. If you’d like to see a similar probability / counting problem check out our exploration of the “Snapchat problem” from a few years ago:
Exploring the “Shapchat Problem” with my kids
Last month I learned about a terrific online random walk program to share with kids:
Here’s our project with that program:
Sharing a great random walk program with kids
Last week Dirk Brockmann shared a new program:
Today I showed the boys Brockmann’s original random walk program followed by the new “Anomalous Itinerary” program to see what the boys would think about them.
My older son played with the programs first. Here are his thoughts looking at the original program – he thought the this random walk would be a good description of a particle moving through air:
And here are his thoughts on the new program. One thing that I found really interesting is that he found it difficult to describe the difference between what he was seeing here vs the prior random walk program:
Next up was my younger son. Here are his thoughts on the original program – he thought this random walk would be a good description of how a chipmunk moves.
Here are his thoughts on the new program. He initially thought this was the same as the Gaussian random walk program, but was eventually able to describe the difference:
These programs are definitely fun to share with kids. The “Lévy Flight” paths are definitely not intuitive and very different from the Gaussian random walks. It is really interesting to hear kids trying to find the words to describe what they are seeing.
Yesterday we did an introductory project for kids on Futility Closet’s Paradox of the Second Ace:
Here’s that project:
Introducing the boys to Futility Closet’s Paradox of the Second Ace
Today we continued the project and calculated the two probabilities in the “paradox.” These calculations are pretty challenging ones for kids, but even with the counting challenges, this was a really fun project.
I started by reminding them of the problem and getting their thoughts from yesterday:
Now we calculated the probability of having a second ace given that you have at least one ace. It took a while to find the right counting ideas, but once they did the calculation went pretty quickly. The counting technique that we used here was case by case counting:
Next we moved to the 2nd problem -> If you have the Ace of Spades, what is the probability that you have more than one ace? The counting technique that we used here was complimentary counting:
Finally, I asked the boys to reflect on the problem – was it still a “paradox” in their minds or did it make a bit more sense now that we worked through it?
I really loved talking through this problem with the boys – thanks to John Cook for sharing it and to Futility Closet for writing about it originally!
I saw this tweet from John Cook’s Probability Fact earlier this week:
My reaction was that it would be fun to turn this into a project for kids, but this one would need a little introduction since conditional probability can be incredibly non-intuitive. During the week I came up with a plan, and we began to look at the problem this morning.
Here’s the introduction – I asked the boys to give their initial reaction to the seeming paradox:
Next we looked at an example that is slightly easier to digest -> rolling two dice and asking “do you have at least one 6?”
My younger son had a little trouble with the conditional probability, so I’m happy that we took this introductory path:
Next we moved to a slightly more difficult problem -> rolling 3 distinct dice. I used a 6x6x6 Rubik’s cube to represent the 216 states. To start, I asked the boys to count the number of states that had at least one six. Their approach to counting those 91 states was really fascinating:
Finally, we looked at the analogy to the 2nd ace paradox in our setting. So, if you have “at least one 6” what is the chance that you have more than one six, and if you have “a six on a specific die” what is the chance that you have more than one six?
Again, my younger son had a little trouble understanding how the cube represented the various rolls, but being able to hold the cube and see the states helped him get past that trouble:
Tomorrow we’ll move on to studying the paradox with the playing cards. Hopefully today’s introduction helped the boys understand
I saw a neat problem from John Allen Paulos earlier in the week:
Today my older son was working on a different math project, so I thought I’d use Paulos’s problem for a nice project with my younger son.
I started by introducing the problem (and forgetting to zoom out after introducing it – sorry about the middle 3 min of this video . . . .). Despite the filming goof, you’ll see my son head down a path that illustrates a common counting mistake.
Now we found that our probabilities didn’t add up to 1, so we tried to found out where we went wrong. Fortunately, my son was able to track down the error.
The next part of the project was discussing the expected amount we’d win playing this game. I didn’t define “expected value” but my son was able to come up with a good way of thinking about the concept.
Finally, we went to the computer to write a little program in Mathematica. This part of the project turned out to be a nice lesson in both simulations and in statistics.