Bayes’ theorem — Edexcel A-Level Statistics
Test yourself on Bayes’ theorem with PEARSON EDEXCEL A-Level practice questions.
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Bayes’ theorem explained
Conditional probability gives the chance of one event given that another has occurred, written P(A|B) = P(A∩B)/P(B) when P(B) > 0.
Read the full explanation
Bayes’ theorem reverses the conditioning: P(A|B) = P(B|A)P(A)/P(B). For up to three events, partition the sample space into mutually exclusive, exhaustive events A₁, A₂, A₃ with prior probabilities P(Aᵢ). Then P(B) = Σ P(B|Aᵢ)P(Aᵢ), and P(Aᵢ|B) = P(B|Aᵢ)P(Aᵢ)/P(B). A tree diagram organises this: first branches show the Aᵢ with their priors, second branches show B or B′ with conditional probabilities, and multiplying along a path gives a joint probability. For example, if factory A₁ makes 40% of items with 2% defective, A₂ makes 35% with 3% defective and A₃ makes 25% with 5% defective, then P(defective) = 0.4×0.02 + 0.35×0.03 + 0.25×0.05 = 0.031, and P(A₃|defective) = 0.25×0.05/0.031 ≈ 0.403.
Your focus
- Calculate conditional probabilities using the definition P(A|B) = P(A∩B)/P(B).
- Apply Bayes’ theorem to find posterior probabilities for up to three events.
- Construct and interpret a tree diagram to organise conditional and joint probabilities.
Bayes’ theorem exam tips
Marking Points
- State the definition of conditional probability: P(A|B) = P(A∩B)/P(B), valid when P(B) > 0.
- State Bayes’ theorem in the form P(A|B) = P(B|A)P(A)/P(B) and identify the prior, likelihood and posterior.
- For a partition A₁, A₂, A₃, use the law of total probability P(B) = P(B|A₁)P(A₁) + P(B|A₂)P(A₂) + P(B|A₃)P(A₃).
- Construct a tree diagram with first-stage branches for the Aᵢ and second-stage branches for B and B′, labelling each branch with its probability.
- Multiply probabilities along a path to obtain a joint probability such as P(Aᵢ∩B) = P(B|Aᵢ)P(Aᵢ).
- Divide the relevant joint probability by P(B) to obtain the required posterior probability P(Aᵢ|B).
- Check that the partition is mutually exclusive and exhaustive and that probabilities on each set of branches sum to 1.
Examiner Tips
- 💡Draw the tree diagram before calculating; label every branch with its probability and check that branches from the same node sum to 1.
- 💡Write down the general formula you are using, then substitute values, so the method is clear even if an arithmetic slip occurs.
- 💡Check whether the question asks for a conditional probability or a joint probability; the wording 'given that' signals a conditional probability.
- 💡Keep probabilities as fractions or decimals consistently and round only at the final stage.
Common Mistakes
- Confusing P(A|B) with P(B|A). Correction: these are generally different; use Bayes’ theorem to reverse the conditioning, for example P(A|B) = P(B|A)P(A)/P(B).
- Using P(B) = P(B|A₁)P(A₁) only, ignoring the other events in the partition. Correction: sum over all events in the partition, P(B) = Σ P(B|Aᵢ)P(Aᵢ).
- Multiplying along a tree path but then forgetting to divide by the total probability of the observed event. Correction: the posterior is the joint probability of the path divided by the sum of all paths leading to that observation.
- Treating events in a partition as independent. Correction: events in a partition are mutually exclusive, so they cannot be independent unless one has probability 0.