chance
16 theorems carry this thread across the degree.
The law of total probability
Probability of an event via a partition of the sample space.
Bayes' theorem
How to invert conditional probabilities.
Linearity of expectation
Expectation of a sum is the sum of expectations, dependence notwithstanding.
Markov's inequality
A tail bound from the mean alone for non-negative variables.
Chebyshev's inequality
A tail bound from the variance.
The weak law of large numbers
Sample means converge in probability to the expectation.
Carathéodory's extension theorem
A premeasure extends to a genuine measure.
The monotone convergence theorem
Integrals commute with increasing limits of non-negative functions.
Fatou's lemma
The integral of a liminf is at most the liminf of integrals.
The dominated convergence theorem
A dominated pointwise limit may be integrated term by term.
The Fubini–Tonelli theorem
When iterated integrals may be exchanged.
The Borel–Cantelli lemmas
When infinitely many events occur, almost surely or not.
The strong law of large numbers
Sample means converge almost surely to the mean.
The central limit theorem
Sums of independent variables are asymptotically Gaussian.
Conditional expectation as projection
Conditioning is orthogonal projection in L2.
The martingale convergence theorem
A bounded martingale converges almost surely.