Probability

Introduction When we reason qualitatively, we often rely on our intuition. However, intuition is often loaded with certain meta-biases that cloud our judgment; one of these biases comes into play when we think about “local” and “global” statements. What is a local or a global statement? One way of distinguishing between them is how many conditions we must guarantee hold before we can talk about the statement - so this terminology is relative. A local statement is a statement that uses many more conditions (a higher amount of specificity) than a global statement. For example, any statement about a certain group of people in the past few years is a local statement relative to a statement that is about all species that have ever existed on the earth. ...

As one can easily guess, this blog is about a mental model of thought. I chose to write about it since I feel that introspection about ways of thinking (and consequently what “thinking before doing something” means) is greatly lacking among people, and that it is critical to make better decisions (and realizing when there is no “better” decision). I won’t bore you with the philosophical details, so I’ll approach it from a probabilistic perspective, which is closer to what I personally choose to think in terms of. A word of caution: I will sometimes oversimplify things in order to drive home the point, but sometimes it might seem to contradict with what I say in the later parts of this post. The key here is context, and if you keep track of it, things will make more sense. To understand probability concepts that I mention here in a bit more detail, I recommend reading my post on probability. If you don’t understand/don’t want to go through the math examples here, don’t worry - I’ll intersperse it with a general idea of what we are trying to do, so looking for those explanations should help. Wherever you find something that’s not something you already know, you should probably just make a note of it and go ahead (and read more about it later). If it is still not clear, you can let me know and I’ll try to clarify that part (for you and the other readers of this post). Or even by just looking up the thing on your favorite search engine/LLM, you’ll likely learn a lot, even if in a less pointed manner. ...

This post was originally written on Codeforces; relevant discussion can be found here. Recently someone asked me to explain how to solve a couple of problems which went like this: “Find the expected time before XYZ happens”. Note that here the time to completion is a random variable, and in probability theory, such random variables are called “stopping times” for obvious reasons. It turned out that these problems were solvable using something called martingales which are random processes with nice invariants and a ton of properties. ...