Two events are called independent if knowledge that one event has happened does not provide any information about whether or not the other has also happened. The formal definition is:

Definition

Two events A and B are called independent if

\[ P(A \textbf{ and } B) = P(A) \times P(B) \]

The table below shows the joint probabilities in an artificial examples about school children.

Joint Probabilities
Mathematical performance
    Poor     Satisfactory Above average Marginal
Underweight 0.0225 0.1125 0.0150 0.1500
Normal 0.0825 0.4125 0.0550 0.5500
Overweight 0.0300 0.1500 0.0200 0.2000
Obese 0.0150 0.0750 0.0100 0.1000
Marginal 0.1500 0.7500 0.1000 1.0000

The mathematical performance categories are independent of the weight categories since

\[ P(Underweight \textbf{ and } Above \text{ } average) = P(Underweight) \times P(Above \text{ } average) \]

and similarly for the other performance and weight categories. All this can be summarised by saying

Weight and mathematical performance are independent

Independence and conditional probabilities

Since \(P(A \textbf{ and } B) = P(A \mid B) \times P(B) \) from the definition of conditional probability, if two events are independent,

\[ P(A \mid B) = P(A) \]

and similarly

\[ P(B \mid A) = P(B) \]

Independence means that knowledge that one event has happened provides no information about whether the other has also happened.

This can be seen in the following conditional probabilities for performance, given weight,

Conditional Probabilities
Mathematical performance
    Poor     Satisfactory Above average Total
Underweight 0.15 0.75 0.10 1.0
Normal 0.15 0.75 0.10 1.0
Overweight 0.15 0.75 0.10 1.0
Obese 0.15 0.75 0.10 1.0

Independence is also evident in the conditional Venn diagram on the left below, but not for the variables on the right.