Relative Frequency and Probability Distributions

This lesson covers relative frequency in more detail and introduces probability distributions — a way of listing all possible outcomes of an experiment alongside their probabilities. These topics are part of the AQA GCSE Mathematics specification and build on the earlier lessons on experimental probability and expected outcomes.

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Relative Frequency Recap

Relative frequency is the proportion of times an event occurs in an experiment:

Relative frequency = frequency of event / total number of trials

Relative frequency provides an estimate of the true probability. It is especially useful when outcomes are not equally likely.

Worked Example 1

A biased spinner is spun 200 times. The results are:

Colour	Frequency
Red	48
Blue	72
Green	80
Total	200

Calculate the relative frequency for each colour:

Colour	Frequency	Relative Frequency
Red	48	48/200 = 0.24
Blue	72	72/200 = 0.36
Green	80	80/200 = 0.40
Total	200	1.00

These relative frequencies can be used as estimates of the true probabilities.

Exam Tip: Relative frequencies should always add up to 1 (or very close to 1 with rounding). If they do not, check your arithmetic.

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What Is a Probability Distribution?

A probability distribution is a table (or list) that shows every possible outcome of an experiment alongside its probability. The probabilities must satisfy two conditions:

1. Each probability is between 0 and 1 (inclusive).
2. All the probabilities add up to exactly 1.

Worked Example 2

A biased die has the following probability distribution:

Score	1	2	3	4	5	6
P(score)	0.1	0.15	0.2	0.2	0.15	0.2

Check: 0.1 + 0.15 + 0.2 + 0.2 + 0.15 + 0.2 = 1.00 (valid probability distribution)

Exam Tip: If a question gives you a probability distribution with one value missing, find it by subtracting the sum of the known probabilities from 1.

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Finding a Missing Probability

Worked Example 3

A random variable X has the following probability distribution:

X	1	2	3	4	5
P(X)	0.1	0.25	p	0.2	0.15

Find the value of p.

All probabilities must sum to 1:

0.1 + 0.25 + p + 0.2 + 0.15 = 1

0.7 + p = 1

p = 0.3

Worked Example 4

A spinner has four sections: A, B, C, D.

Section	A	B	C	D
P(section)	3x	2x	x	4x

Find the value of x and hence the probability of each section.

3x + 2x + x + 4x = 1

10x = 1

x = 0.1

Section	A	B	C	D
P(section)	0.3	0.2	0.1	0.4

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Building a Probability Distribution from Data

Worked Example 5

A shop records the number of returns per day over 50 days:

Returns per day	0	1	2	3	4
Number of days	10	15	12	8	5

Build a probability distribution using relative frequency:

Returns per day	0	1	2	3	4
P(returns)	10/50 = 0.2	15/50 = 0.3	12/50 = 0.24	8/50 = 0.16	5/50 = 0.1

Check: 0.2 + 0.3 + 0.24 + 0.16 + 0.1 = 1.00 (valid)

Exam Tip: When building a probability distribution from data, divide each frequency by the total. These are estimates based on the data — with more data, the estimates would become more accurate (law of large numbers).

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Using a Probability Distribution to Find Probabilities

Worked Example 6

Using the probability distribution from Worked Example 2:

Score	1	2	3	4	5	6
P(score)	0.1	0.15	0.2	0.2	0.15	0.2

(a) P(score is even) = P(2) + P(4) + P(6) = 0.15 + 0.2 + 0.2 = 0.55

(b) P(score is greater than 3) = P(4) + P(5) + P(6) = 0.2 + 0.15 + 0.2 = 0.55

(c) P(score is at most 2) = P(1) + P(2) = 0.1 + 0.15 = 0.25

(d) P(score is not 3) = 1 - P(3) = 1 - 0.2 = 0.8

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Combining Probability Distributions with Expected Outcomes

Worked Example 7

Using the same probability distribution, the die is rolled 500 times.

Score	P(score)	Expected frequency
1	0.1	0.1 x 500 = 50
2	0.15	0.15 x 500 = 75
3	0.2	0.2 x 500 = 100
4	0.2	0.2 x 500 = 100
5	0.15	0.15 x 500 = 75
6	0.2	0.2 x 500 = 100
Total	1.00	500

Check: expected frequencies add up to 500 (the total number of trials). This is correct.

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Probability Distributions as Fractions

Probability distributions can also be given as fractions.

Worked Example 8

Outcome	A	B	C	D
P(outcome)	1/6	1/3	1/4	?

Find P(D).

P(D) = 1 - 1/6 - 1/3 - 1/4

Finding a common denominator (12):

= 1 - 2/12 - 4/12 - 3/12

= 1 - 9/12

= 12/12 - 9/12

P(D) = 3/12 = 1/4

Exam Tip: When working with fractions in probability distributions, find a common denominator before adding or subtracting. This reduces errors and makes your working clearer.

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Graphical Representations

Probability distributions can be displayed as bar charts (for discrete data) where the height of each bar represents the probability. The bars should not touch (because the data is discrete).

Key features of a probability bar chart:

• Each bar represents one outcome.
• The height of the bar equals the probability.
• The sum of all bar heights equals 1.

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Summary

• Relative frequency is calculated from observed data: frequency / total trials.
• A probability distribution lists all possible outcomes with their probabilities.
• All probabilities in a distribution must be between 0 and 1 and must sum to 1.
• To find a missing probability, subtract the sum of the known probabilities from 1.
• Probability distributions can be used to calculate expected frequencies by multiplying each probability by the number of trials.
• Relative frequency provides estimated probability distributions from experimental data.