Expected Frequency –

Expected frequency is a straightforward but important probability topic that appears on both Foundation and Higher GCSE Maths papers across AQA, Edexcel and OCR. It links theoretical probability with real-world experiments by predicting how often an event should occur over a number of trials. You also need to compare expected frequencies with actual results to assess fairness. This guide covers the method, worked examples and practice questions. For an overview of every topic, see our complete GCSE Maths topics list.

What Is Expected Frequency?

The expected frequency of an event is the number of times you would expect it to happen based on its probability and the number of trials.

Key Formulas

For example, if the probability of rolling a 6 on a fair dice is 1/6 and you roll the dice 300 times, the expected frequency of rolling a 6 is (1/6) x 300 = 50.

Expected vs Actual

In practice, the actual frequency may differ from the expected frequency due to random variation. The more trials you carry out, the closer the actual frequency should be to the expected frequency.

Fairness

If the actual results differ significantly from the expected results, the object (dice, spinner, coin) may be biased — but you need a large number of trials before drawing this conclusion.

Step-by-Step Method

1. Identify the probability of the event (either given or calculated).
2. Identify the number of trials.
3. Multiply: expected frequency = probability x number of trials.
4. If asked to compare, calculate the difference between expected and actual and comment on whether it suggests bias.

Worked Example 1 — Foundation Level

Question: A fair spinner has 5 equal sections coloured red, blue, green, yellow and white. The spinner is spun 200 times. How many times would you expect it to land on blue?

Working:

P(blue) = 1/5 = 0.2.

Expected frequency = 0.2 x 200 = 40 times.

Answer: You would expect it to land on blue 40 times.

Worked Example 2 — Higher Level

Question: A biased coin has P(heads) = 0.6. The coin is flipped 250 times. (a) Calculate the expected number of heads. (b) The coin actually lands on heads 170 times. Does this support the given probability?

Working:

(a) Expected heads = 0.6 x 250 = 150.

(b) Actual heads = 170. Difference = 170 − 150 = 20. The actual result is 20 more than expected. This is a moderate difference for 250 trials. The given probability may still be approximately correct — some variation is natural. However, the actual frequency is somewhat higher than expected, so if this pattern continued over more trials, it might suggest P(heads) is slightly higher than 0.6.

Answer: (a) 150 heads expected. (b) The difference of 20 is relatively small for 250 trials, so the result broadly supports the given probability, though more trials would increase confidence.

Worked Example 3 — Exam Style

Question: A dice is rolled 600 times. The table shows the results.

Is the dice fair? Justify your answer.

Working:

If the dice is fair, each number has probability 1/6. Expected frequency for each = 600 x (1/6) = 100.

Actual frequencies are all close to 100 (ranging from 95 to 110). The largest deviation is 10 (number 4: 110 vs 100).

Answer: The dice appears to be fair. All actual frequencies are close to the expected frequency of 100. The small differences are consistent with normal random variation over 600 trials.

Common Mistakes

• Expecting exact results. Expected frequency is a prediction, not a guarantee. Actual results will vary due to chance.
• Concluding bias from few trials. A small number of trials can produce large deviations by chance. Always mention sample size when discussing fairness.
• Forgetting to multiply. Some students state the probability as the answer instead of multiplying by the number of trials.

Exam Tips

• Expected frequency questions are quick marks — just multiply probability by number of trials.
• If asked "Is the dice/coin/spinner fair?", compare actual frequencies with expected and comment on whether differences are within reasonable variation.
• Always state that more trials would give more reliable conclusions when evaluating fairness.
• For related topics, see relative frequency and probability scale and basic probability. For key formulas, visit our GCSE Maths formulas page.

Practice Questions

Q1 (Foundation): A fair dice is rolled 180 times. How many times would you expect to roll a 3?

Q2 (Foundation): The probability of it raining on any given day is 0.3. In a 30-day month, how many rainy days would you expect?

Q3 (Higher): A spinner has P(red) = 0.4. It is spun 500 times and lands on red 230 times. Comment on whether the spinner matches the given probability.

Practise expected frequency questions free on GCSEMathsAI.

Related Topics

• Probability Scale and Basic Probability
• Relative Frequency
• Mutually Exclusive Events

Summary

• Expected frequency = probability x number of trials.
• It predicts how many times an event should occur, but actual results will vary due to random chance.
• To assess fairness, compare actual and expected frequencies. Small differences are normal; large, consistent differences may suggest bias.
• Always note that more trials increase the reliability of any conclusions about fairness.