# Probability Theory/Conditional probability

## Basics and multiplication formula

[edit | edit source]**Definition 3.1 (Conditional probability)**:

Let be a probability space, and let be fixed, *such that *. If is another set, then the **conditional probability** of where already has occurred (or occurs with certainty) is defined as

- .

Using multiplicative notation, we could have written

instead.

This definition is intuitive, since the following lemmata are satisfied:

**Lemma 3.2**:

**Lemma 3.3**:

Each lemma follows directly from the definition and the axioms holding for (definition 2.1).

From these lemmata, we obtain that for each , satisfies the defining axioms of a probability space (definition 2.1).

With this definition, we have the following theorem:

**Theorem 3.4 (Multiplication formula)**:

- ,

where is a probability space and are all in .

**Proof**:

From the definition, we have

for all . Thus, as is an algebra, we obtain by induction:

## Bayes' theorem

[edit | edit source]**Theorem 3.5 (Theorem of the total probability)**:

Let be a probability space, and assume

(note that by using the -notation, we assume that the union is disjoint), where are all contained within . Then

- .

**Proof**:

where we used that the sets are all disjoint, the distributive law of the algebra and .

**Theorem 3.6 (Retarded Bayes' theorem)**:

Let be a probability space and . Then

- .

**Proof**:

- .

This formula may look somewhat abstract, but it actually has a nice geometrical meaning. Suppose we are given two sets , already know , and , and want to compute . The situation is depicted in the following picture:

We know the ratio of the size of to , but what we actually want to know is how compares to . Hence, we change the 'comparitant' by multiplying with , the old reference magnitude, and dividing by , the new reference magnitude.

**Theorem 3.7 (Bayes' theorem)**:

Let be a probability space, and assume

- ,

where are all in . Then for all

- .

**Proof**:

From the basic version of the theorem, we obtain

- .

Using the formula of total probability, we obtain

- .