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How do you find the probability of combined events using diagrams and the probability laws?

Two-way tables, sample space diagrams, tree diagrams and Venn diagrams for up to three events; mutually exclusive and exhaustive events; the addition law and the multiplication law for independent events.

A focused answer to Edexcel GCSE Statistics on combined events, covering two-way tables, sample space diagrams, tree diagrams and Venn diagrams for up to three events, mutually exclusive and exhaustive events, the addition law, and the multiplication law for independent events.

Generated by Claude Opus 4.810 min answer

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  1. What this dot point is asking
  2. Diagrams for outcomes
  3. Mutually exclusive and exhaustive events
  4. The addition law
  5. The multiplication law for independent events

What this dot point is asking

Edexcel codes 3p.07 and 3p.08 require you to represent the outcomes of up to three events using two-way tables, sample space diagrams, tree diagrams and Venn diagrams, to understand mutually exclusive and exhaustive events, and to apply the addition law for mutually exclusive events and the multiplication law for independent events. These diagrams and laws are the workhorses of GCSE probability.

Diagrams for outcomes

Edexcel expects fluency with four representations:

  • A sample space diagram lists every possible outcome systematically (for example a grid of the 3636 outcomes of two dice).
  • A two-way table cross-classifies outcomes by two variables and gives frequencies you can turn into probabilities.
  • A tree diagram shows successive events as branches, with a probability on each branch; you multiply along branches and add between the relevant final outcomes.
  • A Venn diagram shows overlapping sets, making "and", "or" and "not" easy to read off, and supports conditional probability.

Choosing the clearest diagram for the situation is itself a skill: trees suit sequences of events, Venn diagrams suit overlapping categories.

Mutually exclusive and exhaustive events

If events are mutually exclusive, there is no overlap, so they cannot both happen. If a set of events is exhaustive, one of them is certain to occur. Recognising whether events overlap decides which form of the addition law to use.

The addition law

The general law subtracts P(A and B)P(A \text{ and } B) because the overlap would otherwise be counted twice. When the events are mutually exclusive the overlap is zero and the law simplifies. Spotting whether to subtract the overlap is a common exam discriminator.

The multiplication law for independent events

Independence usually arises when an item is replaced (so the second pick faces the same probabilities) or when two separate experiments are run. On a tree diagram you multiply the probabilities along the branches of a path to get the probability of that combined outcome, then add the probabilities of the different paths that satisfy the question.

Exam-style practice questions

Practice questions written in the style of Pearson Edexcel exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

Edexcel 1ST0 20204 marksA bag has 44 red and 66 blue counters. A counter is taken, its colour noted, and it is replaced; then a second counter is taken. (a) Draw a tree diagram for the two picks. (b) Find the probability that both counters are red.
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(a) Each pick has P(red)=410=0.4P(\text{red}) = \frac{4}{10} = 0.4 and P(blue)=610=0.6P(\text{blue}) = \frac{6}{10} = 0.6. The tree has two branches at each of the two stages, with these probabilities (the same at both stages because the counter is replaced).

(b) The picks are independent (with replacement), so multiply along the branches: P(red and red)=0.4×0.4=0.16P(\text{red and red}) = 0.4 \times 0.4 = 0.16.

Markers reward a correct tree with the probabilities, recognising independence from replacement, and multiplying to get 0.160.16.

Edexcel 1ST0 20214 marksIn a class, P(studies French)=0.5P(\text{studies French}) = 0.5, P(studies German)=0.3P(\text{studies German}) = 0.3, and P(studies both)=0.1P(\text{studies both}) = 0.1. (a) Explain why French and German are not mutually exclusive here. (b) Using the addition law, find the probability a student studies French or German (or both).
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(a) Mutually exclusive events cannot both happen. Since P(both)=0.1≠0P(\text{both}) = 0.1 \ne 0, some students study both languages, so the events are not mutually exclusive.

(b) The general addition law is P(F or G)=P(F)+P(G)−P(F and G)=0.5+0.3−0.1=0.7P(F \text{ or } G) = P(F) + P(G) - P(F \text{ and } G) = 0.5 + 0.3 - 0.1 = 0.7.

Markers reward explaining that a non-zero "both" probability means not mutually exclusive, and using the addition law (subtracting the overlap) to get 0.70.7.

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