Sharing problem #9 from Mosteller’s 50 Challenging problems in Probability with kids -> how to play craps!`

Problem #9 from Mosteller’s 50 Challenging Problems in Probability is about the game of craps. The question asks, essentially, does the player of the casino have a better chance of winning the game.

This is both a fun and reasonably difficult problem for kids, but it led to a terrific conversation.

Here’s how I introduced the problem:

Following the introduction, I had the boys solve for the probabilities of an immediate win or loss:

Now we moved on to the harder question – what happens if you roll. say, and 8 on the first roll. How do we find the probability that you win the game in this situation?

One way would be summing an infinite series, but I hoped to introduce the boys to a simpler way of seeing the probability here:

Having solved one of the hard cases exactly in the last video, we moved on to solve the rest of them here:

Finally, we went to Mathematica – not for anything super complicated, just to add up the fractions – so we could find out whether or not the player or the casino had the advantage in the game:

This problem is a great one for kids to explore – it really shows how a systematic approach to problem solving can help you get throw a pretty challenging problem.

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Talking through problem #6 from Mosteller’s 50 Challenging Problems in Probability

We are up to problem #6 from Mosteller’s 50 Challenging Problems in Probability. This one asks about the expected loss in a dice game.

The game is played as follows:

(1) You choose 1 number from 1,2,3,4,5,6.
(2) Three six-sided dice get rolled.
(3) If you match 0 numbers you lose the bet,
(4) If you match 1,2, or 3 numbers you get back your original bet plus your bet times the number of matches.

Here’s what the boys thought about the problem:

We actually made some pretty good progress solving the problem in the last video – here we finished solving it:

Finally, we wrote as short program off screen to explore the problem. I thought I’d corrected the lighting problem, but obviously not, unfortunately. Still, hopefully the computer screen shows up well enough to see.

It is fun to see the boys learning to write short programs for these kinds of problems:

Sharing an idea from “How to Gamble if you Must” with my younger son

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:

Playing Mapmaker -> the Gerrymandering Game

Saw a great tweet from Jordan Ellenberg over the weekend:

Though it would be fun to try . . . and the game arrived today!

Tonight I had the chance to play with my younger son – he didn’t know what gerrymandering was, but we talked a little bit about the idea as we opened the box:

Here’s a quick peek at the game set up:

Finally – here’s what the game looked like at the end and then a short discussion of some of the mathematical ideas for kids that come through in the game. What ideas can help you win? What can you do to cause your opponent some problems?

I really like this game a lot. It is easy to play right out of the box – the directions probably took under 5 min to read and understand. The game is a great way to teach important ideas about gerrymandering and also has great math ideas in it for kids. I highly recommend it – thanks to Jordan Ellenberg for the recommendation.

Playing with Colin Wright’s card puzzle

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.

A simple test of our new skew dice

Last week we bought two packs of skew dice from the dice labs:

Today we decided to try a simple statistical test of the disc. Before we dove into that test, though, I asked the boys to tell me how they thought you might go about testing these dice to see how random they were.

We rolled the dice off camera. We rolled them in groups of 10 with each of us shaking up the dice in a container before each roll. After 12 rounds we had a total of 120 numbers -> here are the results:

Although we didn’t really have enough rolls to make a definitely statement about the dice, I think this was a nice way for the kids to see how a simple statistical test would work. I hope the kids are interested in playing around with these dice a bit more.

Using the Infinite Galaxy puzzle from Nervous System to talk topology with kids

The boys and I spent yesterday working on the new Infinite Galaxy Puzzle from Nervous System:

Having finished it, I thought a project talking about some of the math behind the puzzle would be really fun for the boys.

Since the front cover of the puzzle says that it was inspired by the Möbius strip, I started today’s project talking about that shape:

Next we talked about the puzzle and what geometric / topological properties it has. The interesting mathematical question here is whether or not the puzzle is a Möbius strip?

It turns out the puzzle is projective plane!!

We spent the last part of the project today talking about the projective plane and a few other similar shapes.

Even without any of the math, this new puzzle from Nervous System is a really fun challenge. The mathematical ideas behind the puzzle move it from the “fun puzzle” real to the “blow your mind” realm, though!

I’m so happy to have found one of these puzzles at the Nervous System open house last weekend. What an amazing way to share some introductory ideas from topology with kids!