Relative and expected frequencies

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Cambridge O Level Mathematics (Syllabus D) 4024

Relative and expected frequencies

The big idea: When you cannot count equally likely outcomes, you run an experiment and use what happened. When you know a probability, you can turn it round and predict how often something will happen.

Summary — What This Chapter Covers

Relative frequency = number of times an outcome happens ÷ number of trials. It is an estimate of the probability.
More trials give a more reliable estimate.
Expected frequency = probability × number of trials (or × the size of the population).
Fair means every outcome is equally likely; biased means it is not. Random means every item has an equal chance of being chosen.

1. Relative Frequency as an Estimate of Probability

Syllabus 8.2 · Paper 1 & 2

Probability from an experiment

For a biased spinner or a drawing pin, there are no equally likely outcomes to count. Instead, repeat the experiment many times and work out the relative frequency. As the number of trials grows, the relative frequency settles down towards the true probability, so more trials give a better estimate.

Relative frequency relative frequency = number of times the outcome happens ÷ total number of trials

Worked Example

A four-colour spinner is spun 200 times.

Colour	Red	Blue	Green	Yellow
Frequency	45	38	72	45

(a) Find the relative frequency of green.
(b) Is the spinner fair? Give a reason.

Practice Question 1

Ali drops a drawing pin 50 times and it lands point up 18 times. Ben drops the same pin 300 times and it lands point up 123 times.
(a) Whose results give the better estimate of P(point up)? Give a reason.
(b) Use all the results to estimate P(point up), correct to 3 decimal places.

2. Fair, Biased and Random

Syllabus 8.2 · Paper 1 & 2

The words the questions use

Word	Meaning
Fair	every outcome is equally likely, e.g. a fair dice has P(each score) = 1/6
Biased	some outcomes are more likely than others
Random	chosen so that every item has the same chance; there is no way to predict the result

An experiment can only suggest bias. A fair dice can roll more sixes than expected by chance, especially in a small number of throws, so say “the results suggest” and refer to the number of trials.

3. Expected Frequency

Syllabus 8.2 · Paper 1 & 2

Predicting how many

The expected frequency is the number of times you expect an outcome in a given number of trials. Use it to estimate how many people in a population have a property. It is an estimate, not a guarantee, and it does not have to be a whole number.

Expected frequency expected frequency = probability × number of trials

Worked Example

(a) A fair dice is rolled 120 times. How many sixes would you expect?
(b) The probability that a student at a school is left-handed is 0.12. There are 850 students. Estimate the number who are left-handed.
(c) Use the spinner results from Section 1 to estimate the number of times it lands on green in 500 spins.

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Relative and expected frequencies