Mathematics and Statistics · Ch 16 — Probability

Classical (Mathematical) Definition of Probability

When all the outcomes of a random experiment are equally likely (no outcome is favoured over another — a fair coin, a fair die, a well-shuffled pack), the probability of an event is defined by simple counting.

Note

Classical definition of probability

If a random experiment has $n(S)$ equally likely outcomes, of which $n(A)$ are favourable to event $A$ , then the probability of $A$ is

$P(A) = \frac{n(A)}{n(S)} = \frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}.$

Because $A$ is a subset of $S$ , we always have $0 \le n(A) \le n(S)$ , which gives the fundamental bounds:

0 \le P(A) \le 1.

$P(A) = 0$ means $A$ is the impossible event; $P(A) = 1$ means $A$ is the certain event.
For the whole sample space, $P(S) = \dfrac{n(S)}{n(S)} = 1$ .

Complement rule. Since $A$ and $A'$ are mutually exclusive and exhaustive, $n(A) + n(A') = n(S)$ , so dividing by $n(S)$ :

P(A) + P(A') = 1, \qquad \text{hence} \qquad P(A') = 1 - P(A).

This is one of the most useful results in the whole chapter: whenever the favourable outcomes are hard to count directly but the unfavourable ones are easy, compute $P(A')$ and subtract from $1$ (the "at least one" type of problem is the classic example). …

Definition 8Classical probability

For equally likely outcomes, $P(A) = \dfrac{n(A)}{n(S)}$ , the ratio of favourable outcomes to total outcomes; alway …

Definition 9Complement rule

$P(A') = 1 - P(A)$ , since an event and its complement are mutually exclusive a …

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