Randomness, fairness, and expected outcomes

This is WJEC's introduction to theoretical probability. Foundation tier expects students to identify equally-likely outcomes and compute simple probabilities. Intermediate/Higher applies these to expected frequencies in repeated trials.

What is "fair" / "equally likely"?

An experiment is fair if every outcome has an equal chance of occurring. A standard six-sided dice is fair: each of 1, 2, 3, 4, 5, 6 has probability 1/6. A coin is fair if P(H) = P(T) = 1/2.

If outcomes are equally likely:

P(event) = (number of favourable outcomes) ÷ (total number of outcomes).

Probability scale

Probabilities lie on the scale 0 (impossible) to 1 (certain).

• 0 → impossible
• 1/4 → unlikely
• 1/2 → even chance / 50:50
• 3/4 → likely
• 1 → certain

WJEC asks students to mark a probability on a scale worth 1 mark.

Expected frequency

If P(event) = p and the experiment is repeated n times, the expected number of times the event occurs is n × p.

Example: roll a fair dice 60 times. Expected number of 4s = 60 × 1/6 = 10.

This is expected — actual results will vary, but on average the count tends towards 10 with more trials.

Detecting bias from results

If the observed frequency differs significantly from the expected frequency over many trials, the experiment may be biased. Foundation: "compare 24 sixes in 60 throws (expected 10) — the dice is likely biased."

WJEC exam tip

When asked to comment on whether a coin is fair, always reference the expected count, the observed count, and a clear conclusion: "Expected = n × p; observed = X; difference is large/small relative to n, so coin appears biased / fair."