The derivations of the hypergeometric and binomial distributions follow the same general strategy. In both cases, we first identify all elementary outcomes that are equally likely, then count how many of those outcomes correspond to the event of interest, namely obtaining exactly \( k \) successes. The main difference lies in what constitutes an elementary outcome.

Yesterday, I gave lecture "What makes human behavior interesting to physicists?" (lt. "Kuo fizikams įdomus žmonių elgesys?") to a large group of high school students participating in a project "Student for a day", organized by Vilnius University.

While I do not teaching anything directly related to my research, I have volunteered to do a special introductory lecture and try to explain why physicists are interested in social systems. Initially, I intended to tell students more-or-less about the same things I did when giving a talk at Nacionalinė moksleivių akademija. Yet instead of simply trimming the lecture, I have decided instead to focus on a single example and develop it a bit more. So, in the end I have talked about how "normal" (Gaussian) humans create cities that are rather "peculiar". This lead me to talk about Malthus and Verhulst models of population growth (dedicated blog post should be published soon) and then pivot to preferential attachment as seen in Simon model (I hope to prepare dedicated blog post early next year).

If you are interested, you can take a sneak peek at my slides here (though they are in Lithuanian).

In this TEDx talk shares his scientific journey which lead him to study collective behavior in the beautiful game. Watch it to understand how maths can be used to understood football!

Suppose that a box contains \( N \) balls, of which \( \textcolor{red}{M} \) are red ("success") and \( \textcolor{red}{N-M} \) are of another color ("failure"). We now draw balls with replacement (notably, if we draw without replacement, we would end up with hypergeometric distribution), so that after each draw the selected ball is returned to the box before the next draw. Consequently, each draw is independent, and the probability of success remains constant,

\begin{equation} p=\frac{M}{N}. \end{equation}

Let \( Y \) denote the number of red balls drawn ("successes") in the \( n \) draws ("experiments"). We wish to determine the probability that exactly \( \textcolor{red}{k} \) of the \( n \) (all) draws are successes.

In quite a few statistical physics books, probability density functions (abbr. PDF) are introduced instantaneously by using the Dirac delta function. In my view, this approach is not very intuitive. While reading Heinz's book on mathematical modeling [1], I came across a more intuitive explanation, and I would like to share it here on Physics of Risk.