Relative Frequency
Relative frequency is the number of times an event happened divided by the number of trials, and it estimates a probability from an experiment or from data. This NSW Year 9 Advanced maths subtopic revises the Stage 4 idea (NESA outcome MA4‑PRO‑C‑01) and uses it to make predictions.
You will find relative frequencies from tables and dot plots, predict faulty globes or rainy days, and compare experimental results with theoretical probability. You will decide whose estimate is more reliable, judge whether a die is biased, combine results from two groups, estimate the fish in a dam by tagging, and find how many more successes change a rate. It is part of Year 9 Advanced Probability.
Every question with a fully worked solution.
- Relative Frequency - Video - How To Make a Relative Frequency Distribution Table Watch
Theory
Relative frequency is how often something happened divided by the number of trials. It is an experimental probability: an estimate of the true probability that gets more reliable as the number of trials grows. This Year 9 Advanced maths page revises the earlier idea and extends it to bias, predictions and equations.
The frequency of an outcome is the number of times it happened. Its relative frequency is the frequency divided by the total number of trials. Relative frequencies of all the outcomes add to \(1\).
A theoretical probability says what should happen under fair conditions, e.g. \(\dfrac{1}{6}\) for a six. A relative frequency (experimental probability) says what did happen. The two are rarely exactly equal.
With more trials, the relative frequency usually gets closer to the true probability. So a large experiment gives a better estimate than a small one, and a big difference after many trials suggests bias.
When there is no theoretical probability (rain on a date, a faulty product, a free throw), the relative frequency from past records is the best estimate of the probability.
How to find and use a relative frequency
- Find the total number of trials. Add every frequency in the table, graph or dot plot.
- Add the frequencies of the outcomes you want ("fewer than", "at least", "even").
- Divide and give a fraction or a decimal as asked.
- Predict by multiplying the relative frequency by the new number of trials.
Harder questions
- Missing frequency: total minus the known frequencies. Missing total: frequency \(\div\) relative frequency.
- Is it biased? Compare the relative frequency with the theoretical probability; trust it more when there are many trials.
- What is in the bag? Match the relative frequency to the closest possible fraction, e.g. \(0.62\) with \(8\) counters is close to \(\dfrac{5}{8}\).
- Unknowns: write the relative frequency as a fraction with algebra on top and bottom, set it equal to the value and solve.
| People in car | 1 | 2 | 3 | 4 | 5 or more |
|---|---|---|---|---|---|
| Frequency | 46 | 38 | 20 | 12 | 4 |
Total cars: \(46 + 38 + 20 + 12 + 4 = 120\).
| \(\text{(i) more than } 2\) | \(=\) | \(\dfrac{20 + 12 + 4}{120} = \dfrac{36}{120} = \mathbf{0.3}\) |
| \(\text{(ii) at most } 2\) | \(=\) | \(\dfrac{46 + 38}{120} = \dfrac{84}{120} = \mathbf{0.7}\) |
"At most \(2\)" is the complement of "more than \(2\)": \(0.3 + 0.7 = 1\) \(\checkmark\)
(i) Compare the relative frequency of blue with the theoretical probability.
(ii) What does this suggest?
(iii) Estimate the number of blues in \(1000\) spins.
| \(\text{relative frequency}\) | \(=\) | \(\dfrac{142}{400} = 0.355\) |
| \(\text{theoretical}\) | \(=\) | \(\dfrac{1}{4} = 0.25\) |
(ii) After \(400\) spins the relative frequency should be fairly close to the true probability. \(0.355\) is a long way above \(0.25\), so the spinner is probably biased towards blue.
(iii) Use the relative frequency as the estimate, not \(0.25\).
| \(0.355 \times 1000\) | \(=\) | \(\mathbf{355}\text{ blues}\) |
| Colour | red | white | blue | black |
|---|---|---|---|---|
| Frequency | 37 | \(3x\) | 23 | \(x\) |
The unknown is in a frequency and in the total, so write both with \(x\).
| \(\text{total}\) | \(=\) | \(37 + 3x + 23 + x = 60 + 4x\) |
| \(\dfrac{3x}{60 + 4x}\) | \(=\) | \(0.3\) |
| \(3x\) | \(=\) | \(18 + 1.2x\) |
| \(1.8x\) | \(=\) | \(18\) |
| \(x\) | \(=\) | \(\mathbf{10}\) |
(ii) \(60 + 4(10) = \mathbf{100}\) students. Check: \(\dfrac{30}{100} = 0.3\) \(\checkmark\)
Let \(n\) be the number of extra penalties, all saved. Saves become \(18 + n\) and penalties become \(30 + n\).
| \(\dfrac{18 + n}{30 + n}\) | \(=\) | \(0.7\) |
| \(18 + n\) | \(=\) | \(21 + 0.7n\) |
| \(0.3n\) | \(=\) | \(3\) |
| \(n\) | \(=\) | \(\mathbf{10}\) |
Check: \(\dfrac{28}{40} = 0.7\) \(\checkmark\). Both the top and the bottom of the fraction change, so it takes more than you might guess.
Common pitfalls
Frequently asked questions
What is relative frequency?
Relative frequency is the number of times an outcome happened divided by the total number of trials. It is used as an estimate of the probability.
What is the difference between experimental and theoretical probability?
Theoretical probability is worked out from equally likely outcomes, such as \(\dfrac{1}{6}\) for a six. Experimental probability is the relative frequency from actual trials.
Why do more trials give a better estimate?
In a small number of trials, chance can push the results a long way from the true probability. Over many trials these ups and downs tend to even out, so the relative frequency settles near the true value.
How do you calculate expected frequency?
Multiply the probability or relative frequency by the number of trials. For example, \(0.05\) of \(800\) batteries is \(40\) batteries.
How can you tell if a die is biased?
Roll it many times and compare each relative frequency with \(\dfrac{1}{6} \approx 0.17\). If an outcome is far from this after hundreds of rolls, the die is probably biased.