Empirical vs theoretical probability — the law of large numbers

WJEC examines this every Unit 2 paper at Intermediate or Higher. The headline idea:

As the number of trials increases, the experimental (relative) frequency of an event approaches its theoretical probability.

Theoretical probability

Defined when outcomes are equally likely:

• P(event) = (number of favourable outcomes) ÷ (total outcomes)
• Example: P(rolling a 6 on a fair die) = 1/6 ≈ 0.1667.

Experimental (empirical / relative) frequency

Calculated from data:

• relative frequency = (number of times event occurred) ÷ (total trials)
• Example: 6 sixes in 30 rolls gives 6/30 = 0.2.

After a small number of trials, relative frequency can sit far from theoretical. After many trials, the two converge.

Estimating probability from data (when no theoretical model exists)

Sometimes there is no fair-die symmetry — e.g. probability that a drawing pin lands point-up. Drop it 200 times, count the point-ups, and use the relative frequency as your estimated probability. The bigger the sample, the more reliable the estimate.

Reliability and bias

• A small sample (< 30) gives a rough estimate.
• A biased die or coin will not converge to symmetric values — relative frequency will settle on the true (biased) probability.
• WJEC may ask: "Is the die fair?" — compare the relative frequency to 1/6 across enough trials. A persistent gap of, say, 0.25 vs 0.167 over 600 rolls is evidence of bias.

Expected frequency

Once probability is known (or estimated), expected frequency = probability × number of trials. WJEC question: "If P(red) = 0.4, how many reds in 250 spins?" → 0.4 × 250 = 100.

WJEC exam tip

Always state the formula in words for the M1: "relative frequency = successes ÷ trials". When concluding fairness, reference both the theoretical value and the size of the sample. A short sentence — "the sample of 600 is large, so 0.25 is strong evidence of bias" — secures the SC1 communication mark.