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Theoretical Probability

FoundationHigherAQAEdexcel

Revise Theoretical Probability for GCSE Statistics with this free worksheet and full mark scheme — Foundation and Higher exam-style questions with worked answers for AQA and Edexcel. Theoretical probability is the number of favourable outcomes divided by the total number of equally likely outcomes.

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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

Theoretical probability is calculated from the possible outcomes of an event, without carrying out any trials.

The formula is \(P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of possible outcomes}}\). This assumes all outcomes are equally likely, which is why it works for fair dice, fair coins and well-shuffled cards.

If the outcomes are not equally likely, theoretical probability cannot be used this way — a biased dice requires experimental probability instead. Stating the assumption that the dice or coin is fair is often worth a mark, and is the detail most often omitted.

Revision notes

The formula

\(P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of possible outcomes}}\).

Count the outcomes that give the event, then divide by the total number of possible outcomes. Both counts must cover the same set of possibilities.

The equally likely assumption

This formula assumes every outcome is equally likely.

It therefore applies to fair dice, fair coins and well-shuffled cards. If the object is biased, the outcomes are not equally likely and experimental probability must be used instead.

Stating the assumption

Questions often award a mark for stating that the dice or coin is assumed to be fair.

This is the detail most often omitted. If a question describes an object as biased, theoretical probability calculated this way will be wrong.

Key points

  • Theoretical probability is calculated, not measured.
  • \(P = \frac{\text{favourable}}{\text{total possible}}\).
  • It assumes all outcomes are equally likely.
  • It applies to fair dice and coins.
  • Biased objects need experimental probability.
  • State the fairness assumption when relevant.

Worked examples

Example 1

A fair dice is rolled. Work out the probability of getting an even number. [2 marks]

Working

Favourable outcomes are 2, 4 and 6, so there are 3 out of 6 possible outcomescount the favourable and total outcomes
\(P(\text{even}) = \frac{3}{6} = \frac{1}{2}\)write and simplify the probability

Example 2

A bag contains 5 red and 7 blue counters. Work out the probability of picking a red counter. [2 marks]

Working

Total counters \(= 5 + 7 = 12\)find the total number of outcomes
\(P(\text{red}) = \frac{5}{12}\)write the probability

Example 3

Explain why theoretical probability cannot be used for a biased dice. [2 marks]

Working

Theoretical probability assumes all outcomes are equally likelystate the assumption
but a biased dice makes some outcomes more likely than others, so experimental probability must be used insteadexplain why it fails

Common mistakes

  • Forgetting to find the total number of outcomes.

    The denominator is the total, which may need calculating.

  • Using theoretical probability for a biased object.

    The equally likely assumption does not hold.

  • Not simplifying the fraction.

    Give the answer in its simplest form where possible.

  • Counting outcomes inconsistently.

    Favourable and total must come from the same set of possibilities.

Exam tips

  • Count the total outcomes carefully first.
  • State the fairness assumption when relevant.
  • Simplify fractions where possible.
  • Check the object is not described as biased.

Key terms

Theoretical probability
Probability calculated from possible outcomes.
Favourable outcome
An outcome that gives the event.
Equally likely
Every outcome having the same probability.
Biased
Not equally likely, requiring experimental probability.

Related topics

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