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# 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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