Relative frequency, theoretical probability, the 0–1 scale

OCR Foundation papers always include 1–2 marks on the probability scale and the relationship between relative frequency and theoretical probability.

The probability scale

Probabilities are numbers between 0 and 1 inclusive.

• 0 = impossible.
• 1 = certain.
• 0.5 (or 1/2) = even chance.
• Words: very unlikely (≈ 0.1), unlikely (≈ 0.3), even (0.5), likely (≈ 0.7), very likely (≈ 0.9).

Probabilities are usually written as fractions, decimals, or percentages — but never as ratios (avoid "1 in 6" — write 1/6).

Theoretical probability

Counts equally-likely outcomes:

P(event) = (favourable outcomes) / (total outcomes).

Used when the structure is symmetric (fair coin, fair die).

Experimental / relative frequency

Counts observed outcomes:

Relative frequency = (number of times event occurred) / (total trials).

Used when:

• Theoretical probability isn't known (e.g. probability of a faulty bulb).
• Testing whether a device is fair.

As trials increase, relative frequency converges to the true probability (law of large numbers).

When they agree, when they don't

Situation	Theoretical	Experimental
Fair die, 60 rolls, asking P(6)	1/6 ≈ 0.167	11/60 ≈ 0.183 (slight noise)
Faulty light bulbs from a factory	Unknown	4/100 = 0.04 (estimate from sample)
Suspected biased coin, 1000 flips	(Assume 0.5)	0.55 — significant deviation

Estimating expected counts

Expected count = N × probability.

In a sample of N items, if P(event) = p:

• Expected count of events = Np.
• Useful for both theoretical (fair die expected six count) and experimental probability (predict failures from sample).

OCR mark scheme conventions

• "Mark on the probability scale" → B1 for placing the probability at the correct point.
• Probability written as 1/6: accepted. As 0.167: accepted. As 16.7%: accepted. As "1 in 6": NOT accepted as a probability statement.
• Relative frequency answer: B1 for the fraction; equivalent forms accepted.