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.
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,
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.