Probability fundamentals

The probability scale

All probabilities lie between 0 and 1.

• 0 = impossible
• 1 = certain
• 0.5 = even chance (equally likely)

Probability can be written as a fraction, decimal or percentage. CCEA papers accept any equivalent form unless one is specified.

Theoretical probability

For equally likely outcomes: P(event) = (number of favourable outcomes) ÷ (total number of outcomes).

Example: rolling a 6-sided fair dice, P(even) = 3/6 = 1/2.

Probability of NOT an event

P(not A) = 1 − PA.

This is essential for "at least" problems and is awarded B1/M1 in CCEA.

Mutually exclusive events

Two events are mutually exclusive if they cannot happen at the same time. P(A or B) = PA + PB.

If a set of mutually exclusive events covers all outcomes, their probabilities sum to 1. This is the basis for "find the missing probability" questions.

Expected frequency

If an experiment is repeated n times and an event has probability p, the expected frequency is: Expected frequency = n × p.

Example: Probability of rolling a 6 = 1/6. Expected number of 6s in 60 rolls = 60 × 1/6 = 10.

This is the theoretical expectation — actual results will vary due to randomness.

Relative frequency (experimental probability)

P_experimental(event) = (number of times it happened) ÷ (total trials).

The more trials, the closer relative frequency tends to come to theoretical probability (the law of large numbers).

Common CCEA exam tip

Express probabilities as fractions in simplest form unless told otherwise — leaving 4/8 instead of 1/2 can lose the final A1 mark. Decimals/percentages are equally accepted unless the question specifies "give your answer as a fraction".