Probability of Single Events Notes — Sec 2 Math

The probability scale

$0 \le P(\text{event}) \le 1$

P = 0 means impossible, P = 1 means certain and P = 1 ÷ 2 is an even chance. A probability can never be negative and can never exceed 1.

Theoretical probability

$P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of equally likely outcomes}}$

Valid only when every outcome is equally likely — look for the words “fair”, “unbiased”, “at random”, “well shuffled” or “equal sectors”.

Theoretical probability (set notation)

$P(E) = \frac{n(E)}{n(S)}$

n(E) is the number of outcomes in event E; n(S) is the number of outcomes in the sample space S.

Complement rule

P(A′) = 1 − P(A)

A′ is the event “A does not happen”. Use it for “not” questions, e.g. for a fair die P(not 6) = 1 − 1 ÷ 6 = 5 ÷ 6.

Relative frequency (experimental probability)

$\text{relative frequency} = \frac{\text{number of times the event occurs}}{\text{total number of trials}}$

As the number of trials grows, it stabilises towards the theoretical probability. It is the only way to estimate probability for objects with no symmetry, such as a drawing pin or a biased die.

Spinner with unequal sectors

$P(\text{sector}) = \frac{\text{sector angle}}{360^\circ}$

Use when sectors differ in size and counting sectors is not valid. A 90° sector has probability 90 ÷ 360 = 1 ÷ 4, regardless of how many sectors the wheel has.