A simple & clear explanation of the Gini impurity?

Imagine an experiment with $k$ possible output categories. Category $j$ has a probability of occurrence $p(j|t)$ (where $j=1,..k$)

Reproduce the experiment two times and make these observations:

• the probability of obtaining two identical outputs of category $j$ is $$p^2(j|t)$$
• the probability of obtaining two identical outputs, independently of their category, is: $$\sum\limits_{j=1}^k p^2(j|t)$$
• the probability of obtaining two different outputs is thus: $$1-\sum\limits_{j=1}^k p^2(j|t)$$

That’s it: the Gini impurity is simply the probability of obtaining two different outputs, which is an “impurity measure”.

Remark: another expression of the Gini index is:
$$\sum\limits_{j=1}^k p_j(1-p_j)$$
This is the same quantity:
$$\sum\limits_{j=1}^k p_j(1-p_j) = \left(\sum\limits_{j=1}^k p_j \right) -\left( \sum\limits_{j=1}^k p^2_j \right) = 1 - \sum\limits_{j=1}^k p^2_j$$