On New Year’s eve my older son was playing poker with his friends using 2 decks of cards – see if you can spot the oddity in the picture 🙂
For today’s project we decided to explore some of the probability ideas around playing poker with 2 decks of cards. First we just looked at the possible hands and talked about some potential questions to ask:
For an introductory problem, we looked at the number of ways of getting a Royal Flush and then all types of flushes with a 5 card hand dealt from a single deck of cards:
Now we looked at how regular flushes could happen when dealt from a two decks of cards shuffled together. We also had a good discussion about whether or not it was more likely or less likely to get a flush in the 2 deck situation:
Finally, we went back to Mathematica to take a look at the numbers for a general flush with one and two decks. Here we are lumping all kings of flushes together – regular ones, straight flushes, and Royal flushes are all the same.
Compute the chances of various hands with 2-deck poker is a pretty fun math exercise for kids.
There was yet another absolutely fantastic episode of the Mathematical Objects podcast published last week. I just can’t say enough about the great work that Katie Steckles and Peter Rowlett are doing with this blog:
After listening to the post yesterday I did a little google searching for the game online and thought a project based on the Ox Blocks would be really fun.
First we talked about the game:
After the short introduction the boys played one round of Ox Blocks. Since we didn’t have the specific blocks the game uses, we used a 12 sided die and looked at the rolls mod 3. I think the video will show you (i) how easy the game is to play, and (ii) what a fantastic game it is to play with kids:
Next we rounded up to discuss some of the surprises and strategy from the game:
Finally, we went to Matheamatica to explore some of the statistics from the podcast. Unfortunately iMovie didn’t like something about how the video got recorded, and an hour of trying to fix the problem produced essentially no results.
What we were exploring was the situation that Peter Rowlett described in the podcast. He rolled the cubes 501 times and found roughly 180 “take away” rolls, 160 X’s and 160 O’s. So, how likely was it that he’d have seen one of the rolls come up 180 times in 500 rolls. We found that one of the sides coming up 180 times would happen about 30% of the time if the cubes were fair:
My old son wondered what would have happened had Peter found one of the sides coming up 200 times. Turned out that was far less likely – about 0.25%
I’m sorry that the video for this part of the project broke – it was a fun discussion.
Last week I was really lucky to be able to visit ICERM in Providence and saw some amazing mathematical tiles made by Cherry Arbor Design. Their website is here:
When I got home from that trip I ordered 3 sets of tiles. They arrived today!
Tonight I asked the boys to play around with a set and see what they could make. My younger son chose the Twin Dragon Tiles and played with them for 45 min! Unluckily I had a call that came in at roughly the 2 minute mark of the video below, but we resumed after that call. You can see from the video that he really enjoyed creating all kinds of different shapes:
My older son chose to play around with the Penrose Tiles. These tiles are completely stunning. Here’s his creation and what he had to say about the tiles:
The mathematical tiles and puzzles from Cherry Arbor Design are absolutely beautiful. If you are looking for something fun and math-y to get for someone for a present, definitely check out their selections!
We’ve taken a break from going through Mosteller’s 50 Challenging Problems in Probability, but are back at it today. Problems 11 and 12 are not calculation problems, but rather are discussion problems.
Honestly, at first I wasn’t sure whether or not to even use them, but the discussion turned out to be pretty interesting.
Here’s the first problem:
And the 2nd problem which is similar:
It would be fun to share either of these problems with a classroom – I’m sure the discussion would be great!
I saw a really neat post from Jordan Ellenberg last week:
Jordan Ellenberg’s “The Quarter Circle Game” blog post
It looked like something that the boys might enjoy playing with, so we gave it a shot this morning. Thankfully Ellenberg included some code in his blog post which made it really easy to implement this game in Mathematica. We’ll get to the computer simulations in the last video, but I started out by just explaining the game:
Now we tried to solve the game for a circle of radius 3. I started out with this extra small version of the game to make sure that the kids understood the rules and also to be sure that they knew what “winning position” and “losing position” meant:
Now we moved on to a circle of radius 6. This example was a little harder, but it really helped both kids get to the finish line on understanding how the game worked. I definitely think anyone exploring this game with kids should run through a few small examples first since there are a few potential areas of confusion that probably aren’t obvious to adults.
Here’s how this next case went:
Finally, we moved to the computer program. Roughly speaking the first 3 min of this video are me explaining the code, and the last 5 are playing around with different cases. It was fun to see the kids describe the different patterns they were seeing.
Also, my older son wanted to explore the ratio of winning points to losing points inside the circle. We’ll either tackle that off line or maybe for a project tomorrow.
It is always super fun to be able to share ideas that are both accessible to kids interesting to professional mathematicians. I really like this game since it gives kids a nice opportunity to think through a pretty complex problem and also because the game is so easy to play with on a computer.
Thanks to Jordan Ellenberg for writing up his thoughts (and sharing his code!).
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.
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: