# Ten days of statistics (3) - Probability

## Probability

An experiment is any procedure that can be infinitely repeated and has a limitted set of possible outcomes (sample space). We define an event to be a set of outcomes we interested in. The probability of an event is:

*From here we will consider 2 events $A$ and $B$, in a sample space $S$*

## Events

Let $P(A)$ denotes the probability of event $A$ occurs, $P(A^c)$ denotes the probability of event $A$ not occurs.

A **compound event** $A \cup B$ is an event where either $A$ or $B$ occurs.

$A$ and $B$ is **mutually exclusive** events if they have no events in common. Formally $A \cap B = \varnothing$

An event is said to be **exhaustive** when it equals to $S$. $A$ and $B$ is collectively exhaustive
when their union covers the sample space. Formally $A \cup B = S$ and $P(A \cup B) = 1$

If the outcome of event $A$ has no impact on event $B$, they are considered to be independent. When $A$ and $B$ are independent, $P(A \cup B) = P(A) \times P(B)$

## Conditional probability

This is defined as the probability of an event occurring, assuming that one or more other events have already occurred. Let $P(B|A)$ denotes the probability of $B$ given $A$ occurred. If events $A$ and $B$ are independent. It’s obvious that

If events $A$ and $B$ are not independent, then we must consider the probability that both events occur.

## Bayes’ theorem

Bayes’ theorem is stated mathematically as the following equation

Which also equals to

Where $P(B) \neq 0$

Proof can be found here: https://en.wikipedia.org/wiki/Bayes%27_theorem

Next lesson: Combinatorics