Ryan's Repository of Random Reflections

As many of the people that read this blog probably know, I enjoy inventing math problems, especially problems that are probabilistic in nature, and then working out the math to solve that problem. In this blog post, I will introduce one problem that I solved that I think is particularly elegant: You are randomly walking along the x-axis starting at \( x=0 \). When you are at \( x = n \), you move to \( x = n-1 \) with probability \( p = \frac{n}{n+1} \) and to \( x = n+1 \) with probability \( p = \frac{1}{n+1} \). What is the expected number of steps until you return to \( x = 0 \) for the first time? This problem differs from standard random walk problems because the probability of moving left and right along the x-axis depends on your current location. In this problem, there is a force that is pulling you towards \( x = 0 \), and the further away you get, the more of an effect this force has on you. Show/Hide Solution So how do we go about solving this problem? When...

This semester in my mathematics capstone course, students had the chance to develop mathematical models to describe real world research problems.  I took this as an opportunity to research and develop probabilistic models that can be used to predict the outcome distribution of a baseball at bat.  My hope was that I could apply my findings from this research project to my side project of Beat the Streak.  I learned a lot about mathematical modeling in this class, and explored a variety of techniques for making predictions about baseball at bats.  At the end of the class, we wrote a report that introduces the models we came up with, and compares the quality of the predictions they produce.  I have made this report available here .