A Reflection on Probability and Learning
I’m writing this post fresh off of an exam (on mainly probability), which might reflect on certain opinions. For some context, I have a pretty decent background in mathematics, taking various courses over the years ranging topics from differential equations, linear algebra, multi-variable calculus, etc.
While I don’t think I’m anything close to a mathematician (thought I would love to be), I think that I am fairly well educated in these contexts, and for a majority of those classes I’ve taken I have done well and understood most concepts. As I mentioned earlier, I’m currently taking a course on probability, and while I understand the high-level formulas, and rules/theorems, I seem to be underwhelming when it comes to the application of them.
My initial thoughts as to why this might be comes from the fact that I’ve probably not practiced enough of the application compared to certain other domains in math like calculus. Over the course of my life, I’ve probably solved hundreds if not thousands of derivatives and integrals (and at certain stages, purely for fun). While this is still not a monumental amount, I think it provides a pretty solid foundation for applying the concepts to general problems. With probability, however, I’ve maybe done 1/10th of what I mentioned for calculus.
Another thought as to why, which was backed up by Reddit and other online sources, is that probability is a lot more difficult to grasp for the brain than “regular” math. A lot of calculus, algebra, and more can be gotten around by memorizing formulas, and generalizing problems via application of these formulas. While yes there are some tricks here and there, it usually is pretty straight-forward. With probability, there are theorems and rules, but at least for me, it does not seem to be that straight-forward of an application. I’m not sure if it’s just a lack of practice (so far it seems that way), or that my brain is too calculation-pilled into trying formulas (which doesn’t directly work for probability)
What I’ve realized, is that the way I learn best is through application and trials and actual output. I’ve been recently learning about more complex types of neural networks, and because I’ve applied and iterated on the theory that I learn, my brain feels things clicking. This is why I will challenge myself over this winter break to properly learn probability. I will be reading research papers on probability, and reading the textbook “Probability Theory: The Logic of Science” by E.T Jaynes. Apart from just reading, I will be completing all of the problems. If at any point I get stuck, I might consult online sources or LLMs, but I rather not. One key useful way in which I’ve been using LLMs to learn (further than just concept explanation) is asking them to generate sample questions based on concepts which actually have been very helpful. One could argue though, that I should probably come up with my own questions.
The motivation for this is not just learning (although that’s probably one of the best), but also because I love math. I’ve been loving math ever since in high school when I was practicing for the Math Olympics, but as I’ve grown older I’ve become more appreciative of the language of the universe. My goal is not to know it all (because there will always be a German professor who knows more than me [shoutout Dr. Blume]) but to have a strong foundation of probability given it’s many applications in the world of engineering and machine learning.
Cheers to more learning ~