Probability

Probability is the math of how likely things are, and a lot of engineering intuition rests on it. It tells you how soon random IDs will collide, how many of a thousand servers will be down at once, and why the slowest one percent of requests matters when a page makes fifty calls. The birthday paradox is the classic example: with just 23 people there’s a better than even chance two share a birthday, because the number of pairs grows much faster than the number of people. The same reasoning sizes hash tables and Bloom filters. Bayes’ rule, for updating a belief with new evidence, explains why an accurate test can still be mostly wrong about a rare condition.