Expected frequency is how many times you expect an outcome to happen in a number of trials: expected frequency = probability × number of trials. If a spinner lands on red with probability 0.3 and is spun 200 times, you expect about 60 reds. The same formula works with relative frequency from an experiment, and it can be rearranged to find the number of trials.
Expected frequency
- Expected frequency = probability × number of trials
- Number of trials = expected frequency ÷ probability
- Probability = expected frequency ÷ number of trials
- An expected frequency is an estimate, not a guarantee
Calculating expected frequency
Multiply the probability of the outcome by the number of times the experiment is repeated. With P(win) = 0.15 and 360 games, the expected number of wins is 0.15 × 360 = 54. The answer does not have to be a whole number, though in context it is often rounded.
Using relative frequency
When the true probability is unknown, use a relative frequency from an experiment instead. If a game was won 22% of the time in trials, the estimated number of wins in 450 games is 0.22 × 450 = 99.
Working backwards
Rearrange the formula to find the number of trials: number of trials = expected frequency ÷ probability. If the probability of a faulty item is 0.08 and 36 faulty items are expected, there were 36 ÷ 0.08 = 450 items.
Expected is not guaranteed
An expected frequency is what you would get on average. In a real experiment the actual number will usually be a little different. A large difference may suggest that the probability used was wrong, for example that a dice is biased.
Expected frequency from a probability table
Sometimes the probabilities come from a table, with one value to find first. Use the fact that the probabilities add up to 1 to find the missing one, then multiply it by the number of trials. For example, if P(green) works out to be 0.15 and the spinner is spun 160 times, about 24 greens are expected.
Fractions as probabilities
The probability may be a fraction. For a fair die rolled 270 times, the expected number of sixes is 1/6 × 270 = 45.
Worked examples
Example 1. A spinner has P(red) = 0.15. It is spun 360 times. How many reds are expected?
- 0.15 × 360.
Answer: 54.
Example 2. In trials, a team won 22% of its games. Estimate the number of wins in 450 games.
- Use the relative frequency 0.22.
- 0.22 × 450.
Answer: 99.
Example 3. A fair die is rolled 270 times. How many sixes are expected?
- P(six) = 1/6.
- 1/6 × 270.
Answer: 45.
Example 4. The probability an item is faulty is 0.08. In a batch, 36 faulty items are expected. How many items are in the batch?
- Number of trials = 36 ÷ 0.08.
Answer: 450 items.
Example 5. A coin is flipped 50 times and lands heads 31 times. A student says it is biased. Comment, using expected frequency.
- Expected heads for a fair coin: 0.5 × 50 = 25.
- 31 is higher, but 50 flips is a small number of trials.
Answer: It might be biased, but more trials are needed to be confident.
Common mistakes
- Dividing instead of multiplying. Expected frequency = probability × number of trials.
- Treating the expected value as certain. It is an estimate; actual results vary.
- Using a percentage as a whole number. 22% = 0.22 in the calculation.
- Forgetting to rearrange when finding trials. Number of trials = expected frequency ÷ probability.
- Judging bias from very few trials. More trials give a more reliable conclusion.
How expected frequency is examined
Expected frequency is tested at Foundation and Higher, frequently combined with relative frequency or a probability table. Questions ask you to:
- work out how many times an outcome is expected
- use relative frequency to estimate an expected number
- find the number of trials from an expected frequency
- comment on whether results suggest bias
Practise with the relative frequency worksheet, which includes expected frequency questions with answers, and the GCSE probability engine.
Expected frequency links closely to relative frequency and to basic probability.
Frequently asked questions
How is expected frequency calculated?
Expected frequency = probability × number of trials.
Will the expected frequency happen exactly?
Not necessarily. It is the average you would expect over many repeats.
How do you find the number of trials?
Divide the expected frequency by the probability.