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The Multiplication Rule

FoundationHigherAQAEdexcel

Learn The Multiplication Rule for GCSE Statistics with this free worksheet and full mark scheme — Foundation and Higher exam-style questions with worked answers for AQA and Edexcel. For independent events, the probability of A and B is P(A) times P(B).

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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.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

The multiplication rule finds the probability that two events both occur.

For independent events, \(P(A \text{ and } B) = P(A) \times P(B)\). Neither probability changes, because one event does not affect the other.

For dependent events, the second probability must be adjusted to reflect what has already happened. Taking two counters without replacement from a bag of 5 red and 7 blue gives \(\frac{5}{12} \times \frac{4}{11}\) for two reds — both the numerator and the denominator change, because one red counter has been removed from the bag.

Revision notes

Independent events

\(P(A \text{ and } B) = P(A) \times P(B)\).

Neither probability changes, because the first event does not affect the second. This applies to repeated rolls, coin flips, and picking with replacement.

Dependent events

The second probability must be adjusted for what has already happened.

From 5 red in 12 counters, two reds without replacement gives \(\frac{5}{12} \times \frac{4}{11}\). Both numerator and denominator fall by one, because a red has been removed.

Combining with the addition rule

Many questions require both rules together.

For two counters of different colours, calculate red-then-blue and blue-then-red separately using multiplication, then add the two results because either order satisfies the condition.

Key points

  • The multiplication rule handles 'and' questions.
  • For independent events, multiply unchanged probabilities.
  • For dependent events, adjust the second probability.
  • Both numerator and denominator change.
  • Check for 'without replacement'.
  • Combine with addition when order can vary.

Worked examples

Example 1

A bag has 5 red and 7 blue counters. Two are taken without replacement. Work out the probability both are red. [3 marks]

Working

\(P(\text{first red}) = \frac{5}{12}\)write the first probability
After removing one red, \(P(\text{second red}) = \frac{4}{11}\)adjust for the counter removed
\(\frac{5}{12} \times \frac{4}{11} = \frac{20}{132} = \frac{5}{33}\)multiply and simplify

Example 2

A coin is flipped and a dice rolled. Work out the probability of a head and a six. [2 marks]

Working

The events are independent, so multiply: \(\frac{1}{2} \times \frac{1}{6}\)identify independence and multiply
\(= \frac{1}{12}\)work out the probability

Example 3

Explain why the second probability changes when counters are taken without replacement. [2 marks]

Working

The first counter is not returned, so there is one fewer counter in the bagstate what changes
and if it was the colour being counted, there is also one fewer of that colour, so both numerator and denominator changeexplain both changes

Common mistakes

  • Not adjusting for dependent events.

    Both numerator and denominator usually change.

  • Adjusting only the denominator.

    If the item removed was the colour being counted, the numerator falls too.

  • Adding instead of multiplying.

    'And' uses multiplication.

  • Forgetting to consider both orders.

    When order can vary, calculate each and add.

Exam tips

  • Check for 'with' or 'without replacement'.
  • Write each probability before multiplying.
  • Adjust both parts of the fraction when dependent.
  • Add the orders when either sequence satisfies the condition.

Key terms

Multiplication rule
The rule for finding \(P(A \text{ and } B)\).
Independent
Events not affecting each other's probability.
Dependent
Events where one changes the other's probability.
Without replacement
Not returning the item, making events dependent.

Related topics

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