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Independent Events

FoundationHigherAQAEdexcelOCR

Master independent events for GCSE Maths with structured, exam-style practice. This Foundation and Higher resource covers probability of independent events and includes a complete mark scheme showing the steps examiners reward. Suitable for OCR. Free to download as a PDF. For independent events, multiply the separate probabilities.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR 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 when the outcome of one does not affect the probability of the other. Rolling a dice twice is independent; the first roll tells you nothing about the second.

For independent events, the probability that both happen is the product of their separate probabilities: \(P(A \text{ and } B) = P(A) \times P(B)\).

Drawing counters with replacement keeps events independent, because the bag returns to its original state. Drawing without replacement makes them dependent, since removing a counter changes both the numerator and the denominator for the second draw.

Revision notes

The multiplication rule

For independent events, multiply the probabilities.

The probability of two heads in two coin flips is \(\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\).

With and without replacement

With replacement, the probabilities stay the same for each draw, so the events are independent.

Without replacement, both the count of the item and the total fall, so the second probability changes and the events are dependent.

Recognising independence

Ask whether the first outcome changes the second's probability.

Coin flips, dice rolls and spinner spins are independent. Drawing cards without replacement is not.

Key points

  • Independent events do not affect each other.
  • \(P(A \text{ and } B) = P(A) \times P(B)\).
  • With replacement keeps events independent.
  • Without replacement makes them dependent.
  • Coin flips and dice rolls are independent.
  • Check whether the first outcome changes the second.

Worked examples

Example 1

A coin is flipped twice. Find the probability of two heads.

Working

\[\tfrac{1}{2} \times \tfrac{1}{2}\]the flips are independent, so multiply
\[= \tfrac{1}{4}\]state the probability

Example 2

A bag has \(3\) red of \(10\) counters. One is drawn and replaced, then another. Find the probability both are red.

Working

\[\tfrac{3}{10} \times \tfrac{3}{10}\]replacement keeps the probabilities the same
\[= \tfrac{9}{100}\]multiply the probabilities

Example 3

Explain why drawing without replacement is not independent.

Working

\[\text{The counter is not returned}\]the bag changes
\[\text{So the second probability differs}\]the events are dependent

Common mistakes

  • Adding instead of multiplying.

    For both events to happen, the probabilities multiply.

  • Treating without-replacement draws as independent.

    Removing an item changes the second probability.

  • Keeping the same denominator without replacement.

    Both the count and the total fall for the second draw.

  • Assuming a previous result affects the next.

    For independent events it does not, however unlikely the run has been.

Exam tips

  • Ask whether the first outcome changes the second's probability.
  • Multiply for independent events happening together.
  • Check whether the question says with or without replacement.
  • Leave answers as fractions unless told otherwise.

Key terms

Independent
Events that do not affect each other.
Dependent
Events where one affects the other's probability.
With replacement
Returning an item before the next draw.
Multiplication rule
Multiplying probabilities for events happening together.

Related topics

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