# 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 itโ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.

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.

# The volume of a sphere via Archimedes

Last week I was looking for a geometry project and found a really cool print on Thingiverse made by Steve Portz:

“Archimedes Proof” by Steve Portz on Thingiverse

When I tried to print it last week I ran into some trouble with the supports and just couldn’t get it to work. During the week, though, I got the idea to print the top and bottom separately and then melt them together!!ย  Today we were able to do our project!

We started by talking about the volume of a cylinder. After our Dan Anderson-inspired 3D Printing and Calculus Concepts for Kids project last week, the boys had some interesting ideas about how to find the volume, though they were confused a little bit about stacking up an infinite amount of circles. I love my younger son’s idea to compare the volume of a cylinder to the volume of one of our erasers.

After we finished talking about cylinders we moved on to pyramids. I started this section by reminding them of this old (and really fun) project about some special pyramids:

A neat geometry project inspired by a James Tanton / James Key tweet

That quick review of pyramids led to thinking about the similarities between pyramids and cones. In particular, we talked about with our old pyramids we were stacking squares and with a cone we were stacking circles. That led to the guess that the volume of a cone was 1/3 base times height, just like a pyramid.

Next we looked at the special situation where the height of a cone and a cylinder was the same length as the radius. In this case we see that the formulas simplify a little and that the cone has 1/3 of the volume of the cylinder in this situation.

In the second half of this video I was trying to make the point that to look at a spherical shape that has the same height as its radius, we should look at a half sphere. To say the least, I did not make this point as clear as I would have liked ๐ฆ

Now we moved on to playing with Steve Portz’s creating from Thingiverse. It is so amazing to see this relationship between the volume of a sphere, cone, and cylinder right in front of your eyes!

Here’s the shorter version of the same thing if you aren’t able to watch the whole video:

So, a super fun 3d geometry project. A million thanks to Steve Portz for posting his creating on Thingiverse!!