Bayes' Theorem
The rule for updating a belief after seeing evidence, by combining how well the evidence fits with how plausible the belief was beforehand.
Bayes' theorem states . In words: the updated belief is the likelihood of the evidence times the prior belief, divided by how likely the evidence was overall.
Its most useful lesson is the base-rate trap, and it is worth doing the arithmetic once. A test that is 99% accurate for a disease affecting 1 in 10,000 people returns a positive result. Out of a million people, 100 have the disease and 99 test positive; the 999,900 healthy people produce about 9,999 false positives. So a positive result means roughly 99 in 10,098 — under 1%. The test is excellent and the answer is still "probably not", because the prior was so low.
In machine learning this is the difference between maximum likelihood and MAP estimation: multiplying in a prior is what turns a pure best-fit into a regularised one. L2 regularisation is a Gaussian prior, and Laplace smoothing is a prior that refuses to let any probability reach exactly zero.