Probability theory

The basic theorems of probability can be developed easily from the probability axioms and set theory.

1. The sum of the probabilities of all the elementary events is one.
2. For any arbitrary events A₁ and A₂, the probability of both events is given by the sum of the probabilities for all elementary events in both A₁ and A₂. If the intersection is empty, then the probability is exactly zero.
3. For any arbitrary events A₁ and A₂, the probability of either or both is given by the sum of the probabilities of the two events minus the probability of both.

The formulae below express the same ideas in algebraic terms.

Σi Pr[Ei] = 1

      Pr[A₁ * A₂] = Σ Pr[E_i]

Where E_i is any event in both A₁ and A₂.

      Pr[A₁ + A₂] = Σ Pr[E_i]

Where E_i is any event in either A₁ or A₂.

(in these equations, "+" means "or" and "*" means "and")

The probability of some event happening knowing that another event happened before can be computed using conditional probability.

probability

probability axioms -- probability distribution