A great introductory probability and stats problem I saw from Christopher Long

Christopher Long has written a little bit about some old interview questions the Charlotte Hornets asked when they were looking for quantatative analyist:

Charlote Hornets’ interview question #1

Here’s the question:

Last year the Celtics finished with the 5th worst record in the NBA. Whether it’s two years from now or twenty years from now, what pick are the Celtics more likely to land first, the #1 pick or the #5 pick? Assume that the lottery odds and structure remain the same going forward. Further assume that the Celtics are likely to improve and every year the team will either:

Improve by 3 positions (e.g. they go from 5th worst to 8th worst) with a probability of 0.6
OR fall 2 positions (e.g. they go from 5th worst to 3rd worst) with a probability of 0.4.

The question itself isn’t what caught my eye as a good introductory question, but rather one piece that you have to do to get started.

Here’s the Wikipedia page describing how the NBA lottery works:

The NBA draft Lottery as described by Wikipedia

The page includes this chart showing the probability of a team getting a particular draft pick in the lottery based on their rank at the end of the season (being ranked #1 in this chart means you finished in last place, btw):

So, the neat introductory probability and stats problem is this – based on the description of how the NBA draft lottery works in the Wikipedia article, recreate that chart!

There are a few different ways to do it, and all, I bet, will lead to some neat conversations about probability.

Chess vs. Golf

Rory McIlroy cause a few probably unwanted headlines earlier this week when he said that both Tiger Woods and Phil Michelson were in the “last few holes” of their careers.  Link here if you like blown-out-of-proportion golf gossip:

http://espn.go.com/golf/story/_/id/11509989/rory-mcilroy-defends-comments-tiger-woods-phil-mickelson-says-was-being-complimentary

It made me wonder, though, how old is old when it comes to pro golf?

Then, in an “oh shiny” moment, I was looking at the latest (unofficial) world chess ratings.  It made me wonder which group is older – the current top 20 male chess players or the current top 20 male golfers?  Well, wonder no more:

Age Range    Group 1     Group 2

20 – 24              6                   2

25 – 29              4                   4

30 – 34              4                   5

35 – 39              3                  7

40+                   3                   2

Before you look to see which one is which, have a good think about it.  Current rates as of September 13, 2014 are here:

http://www.2700chess.com/%20

and here:

http://www.owgr.com/ranking%20

Soccer Ball math

Five Triangles posted a neat picture of a standard soccer ball earlier this week:

https://twitter.com/Five_Triangles/statuses/477502488250048512

Seemed like building the shape out of our Zometool set would be a fun exercise after a week of camping, so we gave it a shot this afternoon:

After introducing the problem we started building (off camera).  We’ve done a few other fun exercises with our Zometool set and actually just bought George Hart  and Henri Picciotto’s “Zome Geometry” so the kids are pretty familiar with building structures out of the Zome pieces.  The only trick for this little exercise is that you want to start with edges that are three times the normal length to make the truncation easier.   Also, once you have the icosahedron, it isn’t so obvious where the soccer ball is hiding:

Next we move to the truncation.  Since we started with side lengths that could be easily divided into three parts, truncating the icosahedron isn’t that hard.  It is, however, incredibly interesting to see the “soccer ball” shape emerge from the icosahedron.  The kids were surprised to see that “the pentagons made the hexagons.”  Here’s a peek from about half way through:

 

After we finished building we did a quick wrap up and talked about a few other questions that we could ask about our new shape – things like the number of edges, or number of pentagons.  I also asked them if they thought the Zometool shape was actually the same shape as the soccer ball and was surprised to hear that they thought it wasn’t.  We talked about that for a bit, too:

 

All in all a fun little geometry exercise.   Didn’t want to go into too much depth here since they just got back from a week of camping, but even without the depth they seemed to find all of the building to be really engaging.  Thanks to Five Triangles for the inspiration.

 

Perfect brackets

I’ve been meaning to write about perfect brackets for a while but haven’t been able to figure out what I wanted to say until today.  This morning I woke up to find my son playing a game he calls “dice racing.”  The game involves rolling two 6-sided dice and keeping track of the totals.  The first total you get 42 times “wins.”  You also win since you get a nice little binomial distribution picture when you finish.  This game seemed like a nice lead into a discussion of brackets.  The initial focus for the bracket discussion was exploring some simple 2 and 4 team tournaments:

Next was a 16 team tournament.  The main point here was simply to try to understand how many games you’d have in general.   From there we moved on to talk about the probability of getting selecting a perfect bracket  in a 64 team tournament (assuming that all games were 50/50 coin flips):

Next, with the camera off, we talked through the probabilities of picking a perfect bracket if your chance of picking each game was equal but different from 50%.  In addition to the 9 quintillion number that comes up when you have a 50% chance of picking each game, we looked at what happens when you have a 60%, 70%, and 80% chance of picking the outcome of each game correctly.  The numbers you deal with in this problem are all so large that they are a little difficult for kids to understand, that’s why I wanted to spend time on this part of the perfect bracket problem.  The other thing that I think kids will find surprising is how much the numbers change when you change the chance of picking each game right.  Raising numbers to the 63rd power is not intuitive!

Next we looked at a few other probabilities – 40%, 90%, 69% and 71%.  The first two numbers came from the boys, the second two came from me.  The size of the numbers caused my older son to wonder what would happen if you picked a different bracket every second.   It turns out that if you had a 52% chance of picking each game right and picked one bracket per second it would take about 14 billion years (or about the age of the universe) to pick a perfect bracket!

Finally, some fun numbers from this year’s tournament.  ESPN has close to 11 million people selecting brackets on their website.  They also publish stats at the end of every round.   That allowed us to watch and see what happened to the number of perfect brackets at every stage.  There were a couple of really interesting surprises.  The surprising math behind these specific stats is a little over their heads, but still fun to talk about.

The Mercer upset over Duke was a surprise for two reasons.   First, obviously, any time a 15 seed beats a 2 seed it is a surprise.   Second, only about 2.5% of the 11 million people who entered brackets in the ESPN contest had selected Mercer, but more than 4 times the percent of the people who had perfect brackets at the end of the first day had picked that upset.  Pretty amazing.  But the bigger surprise came next.

Two games later  #10 seed Stanford beat #7 seed New Mexico (the game in between was a 1 vs 16 seed game).  At the time of this game there were about 2,000 remaining perfect brackets in the ESPN contest and when Stanford won the number of perfect brackets dropped to about 60.  Roughly 97% of the perfect brackets were eliminated even though about 40% of the 11 million people in the overall contest had picked Stanford.  Based on the overall numbers you would have expected the number of perfect brackets to go from 2000 to 800, not 2000 to 60.  I have no explanation for why the people with perfect brackets up to that point did not like Stanford, but I’d love to know!

For me, studying perfect brackets is a great way to show kids some fun math.  With so many people enjoying the basketball tournament, I’m sure that many kids will find it interesting to talk about this math.  For me personally, the math behind the perfect brackets is one of the most interesting “real world” math problems I’ve ever worked on.  I hope to be able to write more about it soon.