Classical Definition of Probability

When the sample space $S$ of a random experiment consists of a finite number of equally likely outcomes, the classical (mathematical) definition of probability of an event $E$ is:

Basic properties that follow directly from this definition:

• $0 \le P(E) \le 1$ for every event $E$, since $n(E)$ can range from $0$ (no favourable outcome) up to $n(S)$ (every outcome favourable).
• $P(S) = \dfrac{n(S)}{n(S)} = 1$ — the sure event has probability exactly $1$.
• $P(\emptyset) = \dfrac{0}{n(S)} = 0$ — the impossible event has probability exactly $0$.
• $P(\text{not } E) = 1 - P(E)$ — the probability of an event NOT happening (its complement) is $1$ minus the probability of it happening, since the favourable outcomes for $E$ and for "not $E$" together make up the whole of $S$ with no overlap.

Worked illustration: for a single fair die roll, the event $E =$ "getting a number greater than $4$" has favourable outcomes $\{5, 6\}$, so $n(E) = 2$ against $n(S) = 6$, giving $P(E) = \dfrac{2}{6} = \dfrac{1}{3}$.