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Probability Distributions

HigherHigher tier onlyAQAEdexcel

Learn Probability Distributions for GCSE Statistics with this free worksheet and full mark scheme — Higher tier exam-style questions with worked answers for AQA and Edexcel. A probability distribution lists the possible outcomes of an experiment together with their probabilities.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA and Edexcel specifications. Every worksheet comes with a full mark scheme.

This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.

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WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

A probability distribution lists all the possible outcomes of an event together with their probabilities. This is a Higher-only topic.

Because the outcomes are exhaustive and mutually exclusive, the probabilities must sum to exactly 1. This provides both a check and a method for finding a missing probability by subtraction.

A discrete uniform distribution is one where every outcome has the same probability, such as rolling a fair dice where each face has probability \(\frac{1}{6}\). The expected value of a distribution is found by multiplying each outcome by its probability and adding the results, giving the long-run average outcome.

Revision notes

What a distribution shows

All possible outcomes with their probabilities, usually in a table.

The outcomes are exhaustive and mutually exclusive, so the probabilities sum to exactly 1. A missing probability is found by subtracting the others from 1.

The discrete uniform distribution

Every outcome has the same probability.

Rolling a fair dice gives \(\frac{1}{6}\) for each face. With \(n\) equally likely outcomes each has probability \(\frac{1}{n}\).

Expected value

Multiply each outcome by its probability and add the results.

This gives the long-run average outcome, which need not be a possible outcome itself — the expected score on a fair dice is 3.5, which no face shows.

Key points

  • A distribution lists outcomes and probabilities.
  • The probabilities sum to exactly 1.
  • A missing probability is found by subtraction.
  • A uniform distribution has equal probabilities.
  • Each of n equally likely outcomes has probability 1/n.
  • Expected value multiplies outcomes by probabilities.

Worked examples

Example 1

A distribution has probabilities 0.2, 0.35 and \(x\). Work out \(x\). [2 marks]

Working

The probabilities sum to 1, so \(0.2 + 0.35 + x = 1\)use the sum-to-one property
\(x = 1 - 0.55 = 0.45\)work out the missing probability

Example 2

Outcomes 1, 2 and 3 have probabilities 0.5, 0.3 and 0.2. Work out the expected value. [3 marks]

Working

\(1 \times 0.5 = 0.5\), \(2 \times 0.3 = 0.6\), \(3 \times 0.2 = 0.6\)multiply each outcome by its probability
\(0.5 + 0.6 + 0.6\)add the products
\(= 1.7\)work out the expected value

Example 3

Explain why the expected value need not be a possible outcome. [2 marks]

Working

The expected value is a weighted average of all the outcomesstate what it is
so it can fall between them, such as 3.5 for a fair dice which shows no such facegive an example

Common mistakes

  • Probabilities not summing to 1.

    For an exhaustive distribution they must.

  • Adding the outcomes instead of weighting them.

    Multiply each outcome by its probability first.

  • Rejecting a non-integer expected value.

    It is an average and need not be a possible outcome.

  • Confusing expected value with the most likely outcome.

    They are different — expected value is a weighted average.

Exam tips

  • Check the probabilities sum to 1.
  • Multiply each outcome by its probability before adding.
  • Accept non-integer expected values.
  • Remember this is a Higher-only topic.

Key terms

Probability distribution
All outcomes listed with their probabilities.
Discrete uniform
A distribution with equal probabilities.
Expected value
The long-run average outcome.
Exhaustive
Covering every possible outcome.

Related topics

Written and reviewed against the current AQA and Edexcel specifications. Spotted an error? Let us know.