Probability — Edexcel A-Level Mathematics
Test yourself on Probability with PEARSON EDEXCEL A-Level practice questions.
7 days Premium · Then free forever · No card, no charge
Probability explained
Mutually exclusive events cannot occur together, so for events A and B, P(A ∩ B) = 0 and P(A ∪ B) = P(A) + P(B).
Read the full explanation
Independent events have no influence on each other, so P(A ∩ B) = P(A) × P(B); this multiplication rule is the key test and calculation tool. The two ideas are distinct: mutually exclusive events with non-zero probabilities are not independent, because knowing one occurred changes the other's probability to 0. For example, rolling a fair die, A = 'score 2' and B = 'score 5' are mutually exclusive; A = 'score even' and C = 'score multiple of 3' are independent because P(A ∩ C) = 1/6 and P(A) × P(C) = 1/2 × 1/3 = 1/6. These rules extend to discrete distributions, where probabilities of separate outcomes sum, and to continuous distributions, where probabilities are found from areas under a curve and P(X = x) = 0 for any single value.
3.2 Understand and use conditional probability, including the use of tree diagrams, Venn diagrams, two-way tables. Understand and use the conditional probability formula P(A | B) = P(A ∩ B) / P(B).
Conditional probability is the probability of A given that B has occurred, written P(A | B). The formula P(A | B) = P(A ∩ B) / P(B) requires P(B) > 0 and lets you reverse the conditioning using P(A ∩ B) = P(B) × P(A | B). Tree diagrams show sequential events, with branch probabilities multiplied along a path and added across paths; the second set of branches carries conditional probabilities. Venn diagrams display intersections and unions as regions, so P(A | B) is the proportion of the B region that also lies in A. Two-way tables organise counts by two categories, and conditional probability is a row or column proportion, not a proportion of the whole table. For example, if 60 of 100 students study French, and 24 of those also study German, then P(German | French) = 24/60 = 0.4.
3.3 Modelling with probability, including critiquing assumptions made and the likely effect of more realistic assumptions.
Modelling with probability means representing a real situation with a probability model, then judging how well that model fits. You choose a model, state its assumptions, calculate probabilities and compare predictions with data or with what is plausible. For example, modelling births as independent with P(girl) = 0.5 assumes a constant probability and no family-level or seasonal effects; if the true probability varies, the model misestimates runs of same-sex births. Critiquing assumptions means naming each assumption, explaining why it may fail, and describing the direction of the likely effect on the calculated probability. More realistic assumptions, such as allowing dependence between trials or a changing probability, usually make the model harder to use but more accurate. Assessment rewards clear reasoning about assumptions and their consequences, not just a numerical answer.
Your focus
- Define mutually exclusive and independent events and state the corresponding probability rules.
- Calculate probabilities of unions and intersections using the addition and multiplication rules.
- Explain how the rules apply to discrete and continuous distributions, including why P(X = x) = 0 for continuous variables.
Show all 9 objectives
- Apply the conditional probability formula to calculate P(A | B) and related probabilities.
- Represent and solve conditional probability problems using tree diagrams, Venn diagrams and two-way tables.
- Interpret conditional probability statements correctly in context, distinguishing P(A | B) from P(B | A).
- Construct a probability model for a real situation and state its assumptions.
- Critique assumptions and explain the likely effect of more realistic assumptions on calculated probabilities.
- Refine a model and interpret its predictions in the original context.
Probability exam tips
Marking Points
- States that mutually exclusive events satisfy P(A ∩ B) = 0 and therefore P(A ∪ B) = P(A) + P(B) for two events.
- States that independent events satisfy P(A ∩ B) = P(A) × P(B), and uses this both to test independence and to calculate combined probabilities.
- Distinguishes the two concepts, recognising that mutually exclusive events with non-zero probabilities are dependent, not independent.
- Applies the addition rule to disjoint outcomes of a discrete random variable and the multiplication rule to independent trials.
- Links the rules to continuous distributions by using areas under the probability density function rather than summing point probabilities.
- Interprets a calculated probability in context, including checking whether a proposed independence or exclusivity claim is consistent with the given data.
- States and applies the formula P(A | B) = P(A ∩ B) / P(B), checking that P(B) > 0.
- Rearranges the formula to find P(A ∩ B) = P(B) × P(A | B) or P(B | A) = P(A ∩ B) / P(A).
- Constructs and interprets tree diagrams, multiplying along branches and adding across branches to obtain total probabilities.
- Uses Venn diagrams to identify P(A ∩ B), P(A ∪ B) and conditional probabilities from region areas or counts.
- Extracts conditional probabilities from two-way tables by dividing within the appropriate row or column total.
- Distinguishes P(A | B) from P(B | A) and from P(A ∩ B) in context.
- States the assumptions of a probability model, such as independence, constant probability, equally likely outcomes or a fixed number of trials.
- Explains why an assumption may not hold in the real context, referring to the situation rather than to the mathematics alone.
- Describes the likely direction of the effect of a more realistic assumption on a calculated probability, for example whether it would increase or decrease it.
- Compares model predictions with observed data or expected values and comments on the adequacy of the model.
- Suggests a specific refinement to the model, such as allowing dependence or a varying probability, and explains its consequence.
- Communicates the modelling cycle: specify, calculate, interpret, validate and refine.
Examiner Tips
- 💡Write the defining equation you are using, such as P(A ∩ B) = P(A) × P(B), before substituting numbers, so the method is clear.
- 💡When a question says 'given that', check whether conditional probability is needed rather than the simple multiplication rule.
- 💡For continuous distributions, sketch the density curve and shade the required area to avoid confusing point probabilities with interval probabilities.
- 💡Label every branch of a tree diagram with a conditional probability and check that branches from the same point sum to 1.
- 💡In a two-way table, write the conditioning total underneath the fraction before calculating, to keep the denominator correct.
- 💡Use the formula in symbols first, then substitute, so that a reversal of the conditioning is visible and avoidable.
- 💡Use the context words in the question when naming assumptions, so the critique is specific rather than generic.
- 💡State the direction of the likely effect, such as 'the probability would be underestimated', to show genuine modelling insight.
- 💡Keep numerical work separate from the critique so that both the calculation and the commentary are clearly visible.
Common Mistakes
- Adding probabilities of independent events to find P(A ∩ B); the correction is to multiply, since P(A ∩ B) = P(A) × P(B) for independent events.
- Treating mutually exclusive events as independent; the correction is to note that if P(A) > 0 and P(B) > 0 and the events are mutually exclusive, then P(A | B) = 0, so they are dependent.
- Assuming P(X = x) > 0 for a continuous random variable; the correction is that a single point has probability 0 and probabilities come from areas under the curve.
- Confusing P(A | B) with P(B | A); the correction is to identify which event is given and to divide by P(B) when finding P(A | B).
- Dividing by the grand total instead of the row or column total when using a two-way table; the correction is to use the total of the conditioning group.
- Multiplying branch probabilities along a path but then forgetting to add the probabilities of all paths that satisfy the event; the correction is to sum across the relevant paths.
- Listing assumptions without explaining their effect; the correction is to state how each assumption changes the calculated probability if it fails.
- Claiming a model is 'wrong' without saying which assumption fails or in which direction the estimate is affected; the correction is to link the criticism to a named assumption.
- Assuming independence in a context where events are clearly linked, such as drawing counters without replacement; the correction is to use conditional probabilities for the second draw.