Mutually exclusive events

We start with a simple definition.

Definition

Two events, \(A\) and \(B\) are called mutually exclusive if they have no outcomes in common. More precisely, using set notation, this is when

\[ A \cap B = \emptyset \]

In other words, \(A\) and \(B\) are called mutually exclusive if they cannot occur at the same time. For example, if \(X\) is a numerical measurement,

Properties of probability

In 1933, the Russian mathematician Kolmogorov published an elegant treatise on 'probability functions'. He reduced the 'theory' of probability to a small number of axioms that hold for all definitions of probability. Anything derived from these axioms also holds, no matter how probability is defined, so we no longer need to distinguish between the different definitions in the last section.

Axioms of probability

  1. P(S) = 1       where S is the sample space
  2. 0 ≤ P(E) ≤ 1        for any event, E
  3. P(A or B) = P(A) + P(B)         if events A and B are mutually exclusive

(For completeness, a second version of Axiom 3 is also needed that applies to an infinite number of mutually exclusive events, but we won't require it in this e-book.)

These axioms are central to probability. The rest of probability theory can be derived from them.