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.
A few weeks ago I ran across a copy of Ben Orlin’s Math with Bad Drawings at a book store:
Last night I asked each kid to pick a chapter in the book to read so that we could talk about those chapters for a project today.
My younger son picked the chapter about dice – hardly a surprise as he’s been fascinated with dice forever! Most of the dice you see in this video can be found at the Dice Lab’s website if you are interested in more information about them. Here’s what my younger son had to say about the chapter and about dice this morning:
My older son picked the chapter on p-values – gulp! This topic is pretty advanced and once that isn’t super easy to explore with kids. But I gave it a shot.
First, here’s what he found interesting:
Next I designed a little experiment on Mathematica. For this experiment I wasn’t using p-values but rather confidence intervals – this was just for simplicity, but was still also not super easy for the boys to understand.
In my experiment, I picked 30 numbers from a normal distribution with mean of 5 and standard deviation of 10, and we looked to see if we could tell (statistically) if the mean of the numbers was greater than 0.
What we found was that roughly 25% of the time, 0 was in the 95% confidence interval of the mean. Also, roughly 2.5% of the time, the lower end of that confidence interval was greater than 5 (so we excluded 5 from the confidence interval roughly 5% of the time!).
Hopefully this little experiment helped the kids understand how you could find “wrong” results every now and then:
I love Ben’s book – definitely a fun read and although it isn’t specifically meant for kids, there are plenty of ideas in the book that can be shared with kids.
Saw a neat tweet from Amy Hogan yesterday:
I found a series that gives the solution, but don’t know a closed form for the solution. Still, though, I thought sharing this problem with kids would be fun – some of the basic ideas you need to start down the road to the solution are accessible to kids. I’m really happy with how the discussion went today.
We started out by reading the problem and I asked the boys a few questions to make sure they understood how the game worked:
Next we started talking about the solution. My younger son had a little bit of a misconception initially, but the problem started to make a bit more sense for him after my older son suggested a different approach.
Still, it was interesting to hear the boys discuss whether the probability of winning in exactly 3 tosses was 1/4 or 1/8.
Now we tried to figure out how likely it would be to win with a string of exactly three heads at the end (so your final four flips would be THHH).
Finally, we tried to count the ways that you could win with final flips of THHHH – so winning by flipping four heads in a row. Their guess in the last video was that there would be 3 cases – but they realized fairly early on that there would be more than 3.
After we finished I showed them my code in Mathematica that calculated the sum that gives the final answer. To 10 decimal places the answer is 0.7112119049.
Thank to Amy Hogan for sharing this problem – even though the exact answer is a little out of reach for young kids, it is still a terrific problem for them to study.
Aperiodical is hosting an “internet math off” right now and lots of interesting math ideas are being shared:
The Big Internet Math Off
The shared by Colin Wright caught my attention yesterday and I wanted to share it with the boys today:
The page for the Edmund Harriss v. Colin Wright Math Off
The idea is easy to play with on your own -> deal out a standard deck of cards (arranged in any order you like) into 13 piles of 4 cards. By picking any card you like (but exactly one card) from each of the 4 piles, can you get a complete 13-card sequence Ace, 2, 3, . . . , Queen, King?
Here’s how I introduced Wright’s puzzle. I started the way he started – when you deal the 13 piles, is it likely that the top card in each pile will form the Ace through King sequence:
Now we moved on to the main problem – can you choose 1 card from each of the 13 piles to get the Ace through King sequence?
As always, it is fascinating to hear how kids think through advanced mathematical ideas. By the end of the discussion here both kids thought that you’d always be able to rearrange the cards to get the right sequence.
Now I had the boys try to find the sequence. Their approach was essentially the so-called “greedy algorithm”. And it worked just fine.
To wrap up, we shuffled the cards again and tried the puzzle a second time. This time it was significantly more difficult to find the Ace through King sequences, but they got there eventually.
They had a few ideas about why their procedure worked, but they both thought that it would be pretty hard to prove that it worked all the time.
I’m always happy to learn about advanced math ideas that are relatively easy to share with kids. Wright’s card puzzle is one that I hope many people see and play around with – it is an amazing idea for kids (and everyone!) to see.
We stumbled on this problem in the book my older son is studying over the summer:
A game involves flipping a fair coin up to 10 times. For each “head” you get 1 point, but if you ever get two “tails” in a row the game ends and you get no points.
(i) What is the probability of finishing the game with a positive score?
(ii) What is the expected win when you play this game?
The problem gave my son some trouble. It took a few days for us to get to working through the problem as a project, but we finally talked through it last night.
Here’s how the conversation went:
(1) First I introduced the problem and my son talked about what he knew. There is a mistake in this part of the project that carries all the way through until the end. The number of winning sequences with 5 “heads” is 6 rather than 2. Sorry for not catching this mistake live.
(2) Next we tried to tackle the part where my son was stuck. His thinking here is a great example of how a kid struggling with a tough math problem thinks.
(3) Now that we made progress on one of the tough cases, we tackled the other two:
(4) Now that we had all of the cases worked out, we moved on to trying to answer the original questions in the problem. He got a little stuck for a minute here, but was able to work through the difficulty. This part, too, is a nice example about how a kid thinks through a tough math problem.
(5) Now we wrote a little Mathematica program to check our answers. We noticed that we were slightly off and found the mistake in the 5 heads case after this video.
I really like this problem. There’s even a secret way that the Fibonacci numbers are hiding in it. I haven’t shown that solution to my son yet, though.