Ten days of statistics (3) - Probability

Posted 2 years ago

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:

P = \frac{\text{Number of interested outcomes}}{\text{Total possible outcomes}}

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

P(B|A) = P(B)

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

P(B|A) = \frac{P(A \cap B)}{P(A)}

Bayes’ theorem

Bayes’ theorem is stated mathematically as the following equation

P(A|B) = \frac{P(B|A) \times P(A)}{P(B)}

Which also equals to

\frac{P(B|A) \times P(A)}{P(B|A) \times P(A) + P(B|A^c) \times P(A^c)}

Where $P(B) \neq 0$

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

Next lesson: Combinatorics