Expected frequency tells you how many times an event should occur if you repeat an experiment a set number of times. The formula is: expected frequency = probability × number of trials. It is a prediction, not a guarantee — real results will often differ from the expected value.
What is the expected frequency formula?
Expected frequency = probability × number of trials
The probability must be written as a decimal or fraction (not a percentage) before multiplying.
Worked example 1: A fair six-sided dice is rolled 120 times. How many times would you expect to roll a 4?
- Probability of rolling a 4: P(4) = 1/6
- Number of trials: 120
- Expected frequency = 1/6 × 120 = 20 times
You would expect to roll a 4 about 20 times. In practice, the actual count after 120 rolls might be 17, 23, or any other nearby value.
How does expected frequency differ from theoretical probability?
| Concept | What it is | Example |
|---|---|---|
| Theoretical probability | A fraction/decimal between 0 and 1 | P(heads) = 0.5 |
| Expected frequency | A count — how many times out of n trials | 0.5 × 200 = 100 heads expected |
| Experimental frequency | What actually happened in an experiment | 97 heads recorded |
The key distinction: probability is the prediction per single trial; expected frequency turns that into a predicted count for a given number of trials.
How do you work out expected frequency when events are not equally likely?
Worked example 2: A biased spinner has the following probabilities:
| Colour | Probability |
|---|---|
| Red | 0.3 |
| Blue | 0.5 |
| Green | 0.2 |
The spinner is spun 80 times. Work out the expected frequency for each colour.
- Red: 0.3 × 80 = 24
- Blue: 0.5 × 80 = 40
- Green: 0.2 × 80 = 16
Check: 24 + 40 + 16 = 80. ✓ The expected frequencies must always sum to the number of trials.
How do you find the number of trials from an expected frequency?
If the expected frequency and the probability are known, rearrange the formula:
Number of trials = expected frequency ÷ probability
Worked example 3: A card is drawn at random from a standard 52-card pack. A student draws a card, replaces it, and repeats. She expects to draw a heart 25 times. How many times did she repeat the experiment?
- P(heart) = 13/52 = 1/4 = 0.25
- Number of trials = 25 ÷ 0.25 = 100 trials
How do expected and experimental frequencies compare?
When an experiment is actually carried out, the experimental frequency (observed count) will rarely match the expected frequency exactly. However, as the number of trials increases, the experimental frequency gets closer and closer to the expected value.
Worked example 4: A fair coin is tossed. Expected frequency of heads = 0.5 × n.
| Number of tosses (n) | Expected heads | Typical actual range |
|---|---|---|
| 10 | 5 | 3 – 7 |
| 100 | 50 | 44 – 56 |
| 1000 | 500 | 480 – 520 |
This narrowing of the gap is the foundation of what you will later study as the law of large numbers.
How do you use expected frequency to test whether a spinner is fair?
If the experimental results are very different from the expected frequencies, the spinner may be biased.
Worked example 5: A spinner should give P(1) = P(2) = P(3) = 1/3. After 300 spins, the results are: 1 appears 80 times, 2 appears 120 times, 3 appears 100 times.
| Outcome | Expected | Observed | Difference |
|---|---|---|---|
| 1 | 100 | 80 | −20 |
| 2 | 100 | 120 | +20 |
| 3 | 100 | 100 | 0 |
Outcome 2 is notably more frequent than expected. This is evidence (though not proof) that the spinner may be biased in favour of 2. At KS3 you are asked to comment on whether results suggest the object is fair, based on how close the experimental and expected frequencies are.
Frequently asked questions
Do expected frequencies have to be whole numbers?
No. Expected frequency = probability × n, and this calculation may give a decimal, such as 13.5. A decimal expected frequency is perfectly valid — it simply means that over many repetitions of n trials, you would average 13.5 occurrences. Round only if the question specifically asks for a whole number.
Why does a fair coin not always land on heads exactly 50% of the time?
Because each toss is a random event. The probability of ½ tells you the long-run tendency, not the guaranteed outcome of any particular batch of tosses. Short runs of a random experiment often look "unfair" by chance. Only with a very large number of trials will the experimental frequency approach the theoretical probability closely.
What is the connection between expected frequency and relative frequency?
Relative frequency = (number of times event occurs) ÷ (number of trials). If you run an experiment and calculate the relative frequency, you get an estimate of the probability. Expected frequency goes in the reverse direction: given a known probability, multiply by n to predict the count.
Can expected frequency be used with dependent events?
The basic formula applies to a single event over n independent trials. If events are dependent (for example, drawing cards without replacement), the probability changes each trial, so you cannot simply multiply one probability by n. You would need to calculate expected frequency separately for each trial or use more advanced methods.
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