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Independent Events

FoundationHigherAQAEdexcel

Revise Independent Events for GCSE Statistics with this free worksheet and full mark scheme — Foundation and Higher exam-style questions with worked answers for AQA and Edexcel. Two events are independent if one happening does not affect the probability of the other.

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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

Two events are independent if the outcome of one does not affect the probability of the other.

Rolling a dice twice gives independent events, because the first roll does not change the second. Taking two counters from a bag without replacement gives dependent events, because removing the first counter changes what remains.

The distinction determines how probabilities are calculated. For independent events, \(P(A \text{ and } B) = P(A) \times P(B)\), with both probabilities unchanged. For dependent events the second probability must be adjusted to reflect what has already happened — both the numerator and the denominator usually change.

Revision notes

Independent events

The outcome of one does not affect the probability of the other.

Rolling a dice twice, flipping a coin twice, or picking from a bag WITH replacement all give independent events. The probability stays the same each time.

Dependent events

The outcome of one changes the probability of the other.

Picking two counters WITHOUT replacement gives dependent events. After removing one counter, both the number of that colour and the total have changed.

Adjusting the second probability

For dependent events, adjust both the numerator and the denominator.

From 5 red in 12 counters, after taking one red there are 4 red in 11. Changing only the denominator is the most frequent error and gives the wrong answer.

Key points

  • Independent events do not affect each other.
  • Dependent events do affect each other.
  • With replacement gives independent events.
  • Without replacement gives dependent events.
  • \(P(A \text{ and } B) = P(A) \times P(B)\) if independent.
  • Adjust numerator and denominator for dependent events.

Worked examples

Example 1

A bag has 5 red and 7 blue counters. One red is removed and not replaced. Work out the probability the next counter is red. [2 marks]

Working

After removing one red there are 4 red counters and 11 counters in totaladjust both the numerator and the denominator
\(P(\text{red}) = \frac{4}{11}\)write the new probability

Example 2

Explain why rolling a dice twice gives independent events. [2 marks]

Working

The result of the first roll does not change the dice in any waystate the reason
so the probability of each outcome on the second roll is exactly the same as on the firstexplain the consequence

Example 3

A coin is flipped twice. Work out the probability of two heads. [2 marks]

Working

The events are independent, so multiply: \(\frac{1}{2} \times \frac{1}{2}\)identify independence and multiply
\(= \frac{1}{4}\)work out the probability

Common mistakes

  • Changing only the denominator for dependent events.

    Both the numerator and denominator usually change.

  • Treating without-replacement problems as independent.

    Removing an item changes the probabilities.

  • Confusing independent with mutually exclusive.

    Independent means one does not affect the other; mutually exclusive means they cannot both happen.

  • Adding when you should multiply.

    Use multiplication for 'and', addition for 'or' with mutually exclusive events.

Exam tips

  • Check for the words 'with' or 'without replacement'.
  • Adjust both numerator and denominator when dependent.
  • Multiply for 'and', add for 'or'.
  • Keep independent and mutually exclusive distinct.

Key terms

Independent
One event not affecting the probability of another.
Dependent
One event changing the probability of another.
With replacement
Returning the item, keeping events independent.
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.