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The Binomial Distribution

HigherHigher tier onlyAQAEdexcel

Practise The Binomial Distribution for GCSE Statistics with this free worksheet and full mark scheme — Higher tier exam-style questions with worked answers for AQA and Edexcel. The binomial distribution models the number of successes in a fixed number of independent trials.

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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

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. This is a Higher-only topic.

Four conditions must hold: a fixed number of trials, each trial having only two outcomes, the trials being independent, and the probability of success being constant throughout.

The probability of exactly \(r\) successes in \(n\) trials is \(P(X = r) = \binom{n}{r} p^r (1-p)^{n-r}\), where \(p\) is the probability of success. The mean of a binomial distribution is \(np\), which gives the expected number of successes directly.

Revision notes

The four conditions

A fixed number of trials, \(n\).

Each trial has only two outcomes, success or failure. The trials are independent. The probability of success, \(p\), is constant throughout. All four must hold for the model to apply.

The formula

\(P(X = r) = \binom{n}{r} p^r (1-p)^{n-r}\).

Here \(\binom{n}{r}\) counts the number of ways \(r\) successes can occur among \(n\) trials, \(p^r\) is the probability of those successes, and \((1-p)^{n-r}\) the probability of the remaining failures.

The mean

Mean \(= np\), the number of trials multiplied by the probability of success.

This gives the expected number of successes. For 20 trials with \(p = 0.3\), the mean is \(20 \times 0.3 = 6\).

Key points

  • The binomial models successes in fixed trials.
  • The number of trials must be fixed.
  • Each trial has two outcomes only.
  • The trials must be independent.
  • The probability of success is constant.
  • The mean is \(np\).

Worked examples

Example 1

A binomial distribution has \(n = 40\) and \(p = 0.25\). Work out the mean. [2 marks]

Working

Mean \(= np = 40 \times 0.25\)substitute into the formula for the mean
\(= 10\)work out the mean

Example 2

State two conditions required for a binomial distribution. [2 marks]

Working

There must be a fixed number of independent trialsgive the first condition
Each trial must have only two outcomes, with a constant probability of successgive the second condition

Example 3

Explain why picking counters without replacement is not binomial. [2 marks]

Working

Removing a counter changes the probability of success for the next pickstate what changes
so the probability is not constant and the trials are not independent, breaking two of the conditionsexplain which conditions fail

Common mistakes

  • Applying the binomial without checking the conditions.

    All four must hold, and questions often test whether they do.

  • Using it for without-replacement problems.

    The probability is not constant, so it does not apply.

  • Forgetting the \(\binom{n}{r}\) term.

    It counts the arrangements and is essential.

  • Confusing \(n\) and \(r\).

    \(n\) is the number of trials, \(r\) the number of successes.

Exam tips

  • Check all four conditions before using the model.
  • Use \(np\) for the mean.
  • Keep \(n\) and \(r\) clearly distinct.
  • Remember this is a Higher-only topic.

Key terms

Binomial distribution
A model for successes in fixed independent trials.
Trial
One repetition with two possible outcomes.
\(p\)
The constant probability of success.
Mean
\(np\) for a binomial distribution.

Related topics

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