PROBABILITY
The mathematical study of uncertainty. While we cannot predict the future of a single event, we can predict the behavior of millions. From coin flips to quantum mechanics, probability is the logic of the universe.
The Law of Large Numbers
Look at the background. We are randomly throwing points. At first, the value is chaos. But as $N$ increases, order emerges. The ratio of points inside the circle inevitably converges to $\pi$ (3.14159...).
Random Variables
A variable whose value is unknown (e.g., a die roll).
Independence
When one event does not affect the likelihood of another.
Combinatorics
Counting the number of ways events can occur.
Expected Value
The average outcome if an experiment is repeated infinitely.
Bayes' Theorem
Updating the probability of a hypothesis (A) as more evidence (B) becomes available.
The Monty Hall Paradox
Why Switch?
When you first picked a door, you had a 1/3 chance of being right. That means there was a 2/3 chance the prize was in the "other" group.
When the host opens a door (which he knows is empty), he doesn't change those odds. The 2/3 chance is now concentrated entirely on the remaining closed door.