Louis La Minh Hoang

In this lesson, my focus was on probability, but not the simple kind where you just count outcomes and divide. This was the more challenging, thinking-heavy probability that requires careful logic, structured reasoning, and sometimes surprisingly tricky calculations.

We started by revising the fundamental ideas, such as sample spaces and events, but quickly moved into more complex situations where outcomes are not independent and simple intuition can easily fail. Many problems required breaking situations into multiple stages and tracking probabilities step by step, making sure nothing was double-counted or missed.

A major part of the lesson involved conditional probability, where the probability of an event depends on another event having already occurred. These problems were difficult because they forced me to rethink the sample space after new information was given. Understanding how the “universe of outcomes” changes was more important than any formula.

We also worked on problems involving combinations of events, such as drawing objects without replacement, arranging outcomes in different orders, or finding probabilities that involved several conditions at once. These questions often looked simple at first but turned out to be deceptive, requiring careful case analysis and strong organizational skills.

Some of the hardest questions involved hidden symmetry or complementary probability, where the easiest way to solve the problem was not to calculate the desired probability directly, but instead to find the probability of the opposite event and subtract from 1. This approach saved time but demanded deep understanding of what outcomes were actually being excluded.

By the end of the lesson, it was clear that probability at this level is less about formulas and more about clear thinking and logical structure. Every assumption matters, every condition changes the problem, and a single oversight can lead to a completely wrong answer. Even though the problems were tough, working through them made probability feel more precise, elegant, and intellectually satisfying.