Relative Frequency and Theoretical Probability Revision Notes | GCSE AQA Higher Maths

1. Two Ways to Measure Chance

Here is a question worth thinking about: if you flip a fair coin 10 times and get 7 heads, does that mean the probability of heads is $\frac{7}{10}$? Or is it still $\frac{1}{2}$? The answer depends on which type of probability you are talking about — and that distinction is at the heart of this lesson.

There are two ways to assign a probability to an event. The first uses logic and symmetry — we call this theoretical probability. The second uses real data from experiments or observations — we call this relative frequency. Both sit on the same 0 to 1 scale, but they come from very different places.

2. The Probability Scale

The language you use to describe a probability must match where it sits on this scale. Examiners award marks for using the correct term — so 'unlikely' for a value close to 0, 'likely' for a value close to 1, and 'even chance' or 'equally likely' for a value of exactly 0.5.

3. Theoretical Probability

Theoretical probability is what we predict before any experiment takes place. It relies on the assumption that all outcomes are equally likely — for example, each face of a fair die has the same chance of landing face up.

For example, the probability of rolling a 3 on a fair six-sided die is $\frac{1}{6}$ because there is 1 favourable outcome (rolling a 3) out of 6 equally likely outcomes. This value does not change no matter how many times you roll the die.

4. Relative Frequency

Now consider a situation where you cannot assume equally likely outcomes — perhaps a drawing pin that could land point-up or point-down, or a biased spinner. Here, theoretical probability cannot be calculated from logic alone. Instead, you run an experiment and record what actually happens.

Relative frequency is always a value between 0 and 1, so it sits on exactly the same probability scale as theoretical probability. The key difference is that it comes from observed data rather than from counting equally likely outcomes.

Worked example: Calculating Relative Frequency from a Table

Priya spins a spinner 50 times. It lands on red 18 times, blue 12 times, and green 20 times. Calculate the relative frequency of landing on red. State whether red is likely, unlikely, or has an even chance.

1. Frequency of red $= 18$ Total trials $= 50$ — Read these values directly from the question. [M1] for correct identification.
2. $$\text{Relative frequency} = \frac{18}{50}$$ — Substitute into the formula. [M1] for correct substitution.
3. $$\frac{18}{50} = \frac{9}{25} = 0.36$$ — Divide numerator and denominator by 2 to simplify, or convert to a decimal. [A1] for correct answer.
4. $0.36 < 0.5$, so red is unlikely (closer to 0 than to 1) — 0.36 sits between 0.25 and 0.5 on the scale — this is 'unlikely'. [A1] for correct language.

Answer: Relative frequency of red $= 0.36$. Red is unlikely.

5. Expected Frequency

Once you have a theoretical probability, you can predict how many times an event should occur over a given number of trials. This prediction is called the expected frequency.

Expected frequency is a prediction, not a guarantee. If you roll a fair die 60 times, the expected frequency of rolling a 4 is $\frac{1}{6} \times 60 = 10$. In practice, you might roll a 4 only 8 times or as many as 13 times — and that is completely normal.

Worked example: Calculating Expected Frequency

A fair coin is flipped 200 times. Calculate the expected frequency of getting tails. (This method is suitable for both calculator-allowed questions and questions without a calculator.)

1. $$P(\text{tails}) = \frac{1}{2}$$ — A fair coin has two equally likely outcomes: heads and tails. [M1]
2. $$\text{Expected frequency} = \frac{1}{2} \times 200$$ — Multiply the probability by the number of trials. [M1]
3. $$\frac{1}{2} \times 200 = 100$$ — Half of 200 is 100. [A1]
4. Expected frequency of tails $= 100$ — This means we predict tails will occur 100 times, but the actual result may differ.

Answer: Expected frequency of tails = 100

6. Relating Relative Frequency to Theoretical Probability

Here is where the two ideas connect. When you run an experiment, the relative frequency you observe will rarely match the theoretical probability exactly — but as the number of trials increases, the relative frequency gets closer and closer to the theoretical probability.

This is a key idea in GCSE Higher probability. With a small number of trials, relative frequency can vary quite a lot from the theoretical value. With a large number of trials, the relative frequency becomes a reliable estimate of the true probability.

Notice the pattern in the table: with only 10 flips, the relative frequency of 0.30 is quite far from 0.50. By 1000 flips, it has settled to 0.503 — very close to the theoretical value. This behaviour is what we expect from any fair, random process.

If the relative frequency from a large number of trials is very different from the theoretical probability, that is evidence the experiment or object may be biased. For instance, if a coin is flipped 1000 times and lands heads 720 times, the relative frequency of 0.72 is so far from 0.5 that we would suspect the coin is not fair.

7. Worked Exam Question: Comparing Relative Frequency and Theoretical Probability

Exam questions on this topic often give you a table of experimental results and ask you to calculate relative frequencies, find expected frequencies, or comment on whether a die or spinner appears to be fair. The worked example below models the full approach.

Worked example: Exam-Style: Biased or Fair?

Aisha rolls a six-sided die 300 times. The table shows her results: 1 appears 42 times, 2 appears 55 times, 3 appears 48 times, 4 appears 61 times, 5 appears 44 times, 6 appears 50 times. (a) Calculate the relative frequency of rolling a 4. (b) Calculate the expected frequency of rolling a 4 if the die were fair. (c) Comment on whether the die appears to be fair.

1. $$\text{Relative frequency of 4} = \frac{61}{300}$$ — Frequency of 4 is 61; total trials is 300. [M1] for correct substitution.
2. $$\frac{61}{300} \approx 0.203$$ — $61 \div 300 = 0.2033...$ Rounded to 3 decimal places: 0.203. [A1]
3. $$P(4) = \frac{1}{6} \quad \text{Expected frequency} = \frac{1}{6} \times 300 = 50$$ — $\frac{1}{6} \times 300 = 50$. If the die were fair, we would expect 4 to appear 50 times. [M1, A1]
4. Observed: 61 times. Expected: 50 times. Difference: 11. — 61 is noticeably higher than 50. With 300 trials, this is a meaningful difference.
5. The die may be biased towards 4, as the relative frequency (0.203) is higher than the theoretical probability ($\frac{1}{6} \approx 0.167$). However, 300 trials may not be enough to be certain. — A good exam answer references both the relative frequency and the theoretical probability, and acknowledges uncertainty. [A1]

Answer: (a) $\frac{61}{300} \approx 0.203$. (b) Expected frequency = 50. (c) The die may be biased towards 4, as 61 > 50, but more trials would give a more reliable conclusion.