Probability theory

Probability theory is the mathematical study of probability.

The basic theorems of probability can be developed from the probability
axioms and set theory. The following assumes that there are only countably
many elementary events (which is not the case for many common probability
distributions, such as the normal distribution).

  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.

     [ \sum_i Pr\left[E_i\right] = 1]

     [Pr\left[A_1 \and A_2 \right] = \sum Pr\left[E_i \right]

OR

     [Pr\left[A_1 \and A_2 \right] = \sum Pr\left[E_i \right]

Where Ei is any event in both A1 and A2.

     [Pr\left[A_1 \or A_2 \right] = \sum Pr\left[E_i \right]

Where Ei is any event in either A1 or A2.

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

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