Before the Super Bowl, someone in our office comes around with a 10×10 grid on which folks pay $10 to write their name on a square. One person can buy many squares if desired. Once all squares are filled with names, the digits 0-9 are assigned randomly on both the horizontal and vertical axes. Each axis represents the last digit of the score of one of the teams. So, if you chose square (4,3) and the score at the end of a quarter in the Super Bowl was 24 to 13, you would win $200. If the final score was 34 to 23, you would win an additional $400. Because of the vagaries of football, not all squares are equally likely to come up. The folks at dataists have created probability matrices based on historical Super Bowl data. My coworker Jon used similar data to create a cool web app.

However, players don’t know which combinations they will be assigned before writing their names on the grid. The only decision they can make is about how many squares to buy. I wanted to see if there was an advantage to purchasing more than one square. Using the dataists’ code, I ran some simulations to see what the optimum choice was in terms of how many squares to purchase. As you can see, after 10,000 simulations, there is no clear relationship between the number of squares purchased and profitability. Note that the graphs show absolute profit, not relative. Also, as we run more and more simulations, the gap between maximum and minimum profits for all squares diminishes, suggesting that the expected net profit is essentially zero regardless of the number of squares chosen as the number of simulations approaches infinity. In conclusion, it seems like Super Bowl pools are fun, but not a good way to reliably make money.


Sometimes, especially when reading news articles, I get the feeling people consider probabilities and odds to be the same thing. For example, here is a Business Insider headline claiming that Nate Silver is predicting “92% odds” of Obama victory. I think what they really mean is that Mr. Silver is predicting a 92% probability of Obama victory. There is a big difference between these two statements! Mathematically, the odds of an event are defined as the probability of the event happening divided by the probability that the event doesn’t happen. So, while a probability can take on any value from 0 to 1 (or, in percentage terms, 0%-100%, the odds can range anywhere from zero to infinity. In fact, when the odds of an event are 1, the probability is only 50%. In the example from above, if the odds of Obama victory were really 92% (=0.92), then the probability of victory would be only 0.48, or 48%. Here are some plots showing the relationship between probability and odds:

Finally, here is another blog post from “Simply Statistics” that illustrates the importance of variance in comparing statistical estimates. The main idea is that if the variance of your estimate is small (ie, that the estimate is very precise), then it could be numerically close to some other comparison value but still be considered “significantly different”.


One of my classmates in my statistics class at Georgetown asked me how I went from doing tropical biology to becoming a software tester. I decided to share my response here because it might be interesting to others who are contemplating a radical change in career direction.

I was convinced throughout college that I wanted to be a tropical field biologist. I studied abroad at a field station in Monteverde, Costa Rica, and after graduation spent almost a year doing research in the Philippines on seed dispersal of a particular understory tree. I even got a publication out of the latter in a German ecology journal. So, I was all set to go for a PhD in bio, but while I was in Manila and Bataan I took some time to reflect on where things were going and I realized I didn’t want to do fieldwork for the rest of my life. It is an amazing experience and I’m glad I had the chance to try it out, but as a career it is a very isolated and difficult lifestyle. You have to spend a lot of time in very remote, uncomfortable, and occasionally dangerous situations to collect data. Experiments often fail, and you have to wait a whole year until you can try again (due to the seasonality of flowering, fruiting, migrating animals, etc). Almost no one pays attention to biology research unless it has a medical application, and most tropical biologists spend their lives documenting the ongoing destruction of species and ecosystems they care for very much but feel powerless to protect. What this means is that in order to follow this path, one must be extremely passionate and dedicated to their discipline. I actually had the good fortune to study under such a person in the Philippines- a man named Leonard Co who in my opinion was a modern-day Linnaeus. Sadly, he was shot about a year ago while doing fieldwork. He was a cheerful, multilingual, botanical genius whose whole life was dedicated to scientific discovery and conservation.

Yet as much as I admired him, I realized I did not want to follow in his footsteps, because I couldn’t imagine myself specializing and focusing so intensely on a single subject. I like learning about a wide variety of things, and since the financial crisis was going on at the same time, I started reading a lot about economics, which I didn’t know much about previously. Furthermore, I had never lived in a big city prior to Manila, and even though I always thought of myself as a rural person, I learned to love the excitement and mental stimulation of the urban environment. By the time I came back to the US, I was totally confused about what I wanted to do career-wise, so I just decided to keep an open mind when I got a job offer to do software testing in Charlottesville. I have enjoyed this work for the past 3 years and am grateful to the folks who decided to take a chance on hiring someone based more on his sense of curiosity than his programming experience. Since I still enjoy learning about both biology and economics (and especially how they sometimes intertwine), I decided to go ahead and try for a masters through night classes in something that seems to be useful in both fields, which is how I ended up studying math and statistics. My career philosophy now is more about just making the most of whatever opportunities come around and trying to always be learning something new rather than having a master plan to save the world. I do hope to someday use my technical skills to help conservation efforts in places like the Philippines and other parts of the tropics, but I imagine it will be more in my personal time than as a career. Indeed, I have continued to visit tropical countries and am always inspired by the beauty of the landscapes and the generosity of the people in places like Colombia, Ecuador, Brazil, and Malaysia. I also feel fortunate to have maintained a connection to environmental issues through my work in that most of my projects have been supporting government (EPA) air quality and energy efficiency programs.