Randomness, fairness, and expected frequency

For a fair experiment, all outcomes are equally likely. P(any single outcome) = 1 ÷ (number of outcomes).

Equally likely outcomes

A fair six-sided dice: P(any face) = 1/6. A fair coin: P(H) = P(T) = 1/2. A spinner with 8 equal sectors: P(any sector) = 1/8.

For a fair pack of 52 cards: P(any specific card) = 1/52; P(red) = 26/52 = 1/2; P(king) = 4/52 = 1/13.

Expected frequency

Expected frequency = probability × number of trials.

Worked example: a fair coin tossed 200 times. Expected number of heads = 1/2 × 200 = 100.

Real vs theoretical frequency

If you actually toss the coin 200 times, you might get 96 heads, not 100. The theoretical expectation is 100; the actual frequency (P5) is what you observe.

Edexcel often phrases:

• "How many times would you expect to roll a 5 in 60 throws?" — expected frequency.
• "Janet rolls a dice 60 times and gets a 5 fifteen times. Is the dice fair?" — comparison of observed to expected.

Bias

A coin is biased if outcomes are not equally likely. The probability of an outcome from a biased experiment must be estimated empirically (P3 — relative frequency).

If observed P(H) ≈ 0.65 over many tosses, the coin is biased toward heads. Expected heads in 200 tosses = 0.65 × 200 = 130 (not 100).

Edexcel exam tip

Read carefully whether the experiment is fair (use theoretical probability) or biased (use given relative frequency).

For "expected" answers, give the calculated number — even if it isn't a whole integer. (You'd expect 16.67 heads from 50 tosses of a 1/3-bias coin; that's the right answer, even though you can't have a fractional head.)

Common Edexcel question pattern

A fair dice is rolled 90 times. (a) How many times would you expect a 6 to appear? (b) The dice was actually rolled and a 6 came up 22 times. Comment on whether the dice is fair.

(a) 1/6 × 90 = 15 times. (b) Observed 22, expected 15. Higher than expected, but small samples vary; insufficient evidence on its own that the dice is biased.