Bob Lochel’s question

[only had 30 min to put this together before running out the door, sorry if it seems disjointed]

Saw this question from Bob Lochel today:

Without giving a direct answer to the question, let me share an experience I had with a different problem I saw on twitter last week. I wrote about the first part of it in the link below since I that part of the problem was great for students learning about probability and statistics:

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

Here’s the next part of the story, and one that relates to Lochel’s question. As a reminder, here’s the question at hand:

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.

On twitter, Ben Dilday and Christopher Long came up with a solution using Markov chains and absorbing states. The math behind that solution is something that I’ve seen but not something that I use in my work. BUT, it is clearly a great way to approach this problem and an important piece of math to know if you find yourself needing to solve problems like this one.

My solution was far less sophisticated and much more brute force – no Markov chains, just a simple random walk that started over after every time the Celtics got the 1st or 5th draft pick.

After 10 billion steps in this walk I found 929,797 times when the Celtics received the 5th pick before the 1st pick, and 865,011 times when the reverse was true. So, roughly 48.2% of the time the Celtics got the 1st pick first in my model, and 51.8% of the time they got the 5th pick first. Almost exactly matching Dilday’s and Long’s numbers.

Dilday and Long have nothing to learn from my solution. In fact, Lochel’s question features an almost perfect description of my solution -> correct but wildly inefficient.

I, however, do have a lot to learn from their solution, and have learned from it already. Learning from better solutions (how ever you want to define “better”) is an incredibly important part of development in math. While being able to get to the answer is nice, growing the number of tools in your mathematical toolkit is even nicer!


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