# What are the Odds?

It’s that time of year again, March Madness. Amid exams, studying and the first inklings of Spring, everyone is transfixed by the NCAA basketball tournament and the proverbial brackets. I had an Orgo exam yesterday so I didn’t get to catch much action, but I’ve been glued to the TV today. Unfortunately, Lehigh did not qualify for the tournament this year (remember when we beat Duke), but nonetheless the tourney provides some time to relax, watch sports, and elicit a little friendly rivalry to see who predicts the best bracket. This year, bragging rights aren’t the only thing on the line. I’m sure you’ve heard about the Warren Buffet challenge; he is offering \$1 billion to anybody who completes a perfect bracket. My bracket was unceremoniously busted after the first game when Dayton upset Ohio St. Much publicity has been made about the probability and statistics behind the bracket, and as an engineer I was intrigued by anything math related and decided to take a closer look at the numbers.

There are 9.2 quintillion possibilities to complete a March Madness bracket, that is, you would have a 1 in 9.2 quintillion chance of a perfect bracket if picking games simply by flipping a coin. To put that in perspective, there is better possibility of winning the powerball lottery (1 in 175 million) TWO consecutive days than choosing a perfect bracket at random. However, by using some assumptions and estimates, the odds can be greatly reduced. For example, a 16 seed has never beaten a 1 seed, and the probability of a 15 seed upsetting a 2 is very unlikely (even though Lehigh did it). The mathematical model to represent these estimates and assumptions is immensely complex, and there is no true answer to the exact odds. Just like predicting a bracket, predicting the true odds are basically impossible. I read this interesting article about the odds and statistics of the tourney, it sites Jeffrey Bergen from DePaul Univ as theorizing that the actual odds of predicting a correct bracket are 1 in 128 billion. Using historical data and trends, this statistics were modeled, and realistic a perfect bracket may never be published. Check out the article here: http://fivethirtyeight.com/features/buffetts-billion-wont-lead-to-a-perfect-bracket/

With the \$1 billion slipping through my grasps, I suppose I’ll have to stick to chemical engineering. This got me thinking, how do the odds of the bracket relate to my chemical engineering curriculum? I immediately thought about a Thermodynamics lecture from earlier in the semester. We were discussing entropy and the idea of microstates. Entropy refers to a measure of a system’s disorder and is proportional to the number of microstates that exist for a system. Based on the 2nd Law of Thermodynamics, the entropy of an isolated system cannot decrease, it can either remain the same or increase. Entropy is an imperative property in thermodynamics and is an extremely challenging concept to grasp and understand. Entropy can be defined as S=k*ln(omega), where k is the Boltzmann constant and “omega” is the number of microstates.

In class, we looked at an interesting concept to show how many microstates existed for a theoretical system. Imagine a square with n number of positions and m number of particles. That is, a 5×5 box would have 25 positions. The number of microstates is defined by the equation #=(n!)/[m!(n-m)!]. So if we had a 2×2 box (n=4 positions) and m=2 particles, there would be 6 microstates. Now, imagine a 10×10 box with 100 positions and 50 particles. The number of microstates is an astounding 10^29. This idea is simply a construct to define microstates, if we had an actual gas with, say 1 mole (6.02×10^23 particles) in a container, the number of microstates is ridiculously large. It’s difficult to think about such an abstract idea, but it shows relation to the basketball bracket. As the number of microstates increases, the entropy increases which means the system has more randomness. The different arrangements of teams that could win games in the tournament and scenarios that could play out can be considered all the microstates. The entropy (randomness) of a system with 9.2 quintillion microstates is considerable. It’s interesting to compare theories and ideas from chemical engineering to everyday events, even March Madness. Enjoy some great basketball and plan what you’ll do with \$1 billion (maybe you’re that 1 in 128 billion…).

### Author: Ben Dunmire

I am a sophomore Chemical Engineering major at Lehigh. On campus, I am the president of Club Baseball and a member of AIChE and NSCS. Outside of school, I enjoy the outdoors, fishing, and any type of athletic activity. You can reach me at bcd216@lehigh.edu.