# Recent Posts

### Simpson’s Paradox: A Tale of Two Carpenters

Once upon a time, in a small town, there were two carpenters: Bob and Tom. Bob was an honest, hardworking carpenter with a good reputation. Tom was a somewhat incompetent carpenter with shady business practices. At one point in time, Bob and Tom had each completed 100 carpentry jobs. Bob’s customers were satisfied with 90 out of his 100 jobs. Tom’s customers were satisfied with 95 out of his 100 jobs.

### The Power of Simulation

I’m reading a book that deals with the history of probability theory (Classic Problems of Probability by Prakash Gorroochurn). It is interesting to look back from a modern perspective and realize how many heated arguments could have been resolved in a few minutes if the participants would have had access to a computer. For example, Chapter 1 deals with Cardano and how he (understandably) confused probability with expectation. If you roll a die 3 times, the expectation value of the number of sixes is 0.

### Average Logarithms and the Geometric Mean

I recently came across a surprising connection between the geometric mean and logarithms. At least, it was surprising to me, but isn’t that surprising once you see the derivation (a lot of things in math are this way). The connection is this. Take a set of observations $y_1, y_2, \dots, y_n$. Assume that these are all positive numbers. Then $\langle \log(y) \rangle = \log(G).$ Here the angle brackets indicate the mean (expectation value), and $G$ is the geometric mean of the observations.

In my last post, I discussed the geometric mean and how it relates to the more familiar arithmetic mean. I mentioned that the geometric mean is often useful for estimation in physics. Lawrence Weinstein, in his book Guesstimation 2.0, gives a mental algorithm for approximating the geometric mean. Given two numbers in scientific notation $a \times 10^x \quad \text{and}\quad b \times 10^y,$ where the coefficients $a$ and $b$ are both between 1 and 9.