Bayes Rule

Bayes rule is $$P \left( A | B \right) = \frac{P \left( A \cap B \right)}{P \left( B \right)}$$


There are 5 marbles in a jar: 2 are blue, 3 are red. You pick a marble randomly. If it is blue, you earn \$2, otherwise you lose \$1.

A blue marble has a black dot on it while the other one does not.

Two of the red marbles are marked with a dot.