Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Combined Events & Tree Diagrams

In one sentence. Multiply along branches, add between them — with and without replacement, and the 'at least one' shortcut that saves whole minutes.

Probability — one of 3 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Both tiers.

This module ends with Christiaan Huygens. Learn Combined Events & Tree Diagrams, teach it, then defend it against Christiaan Huygens.

Where this sits in the specification

Specification 4MA1, 6.3 Probability — combined events, tree diagrams. Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 6.3 — Probability — combined events, tree diagrams

1 · Understand it

What it is

A tree diagram maps a two-stage experiment: each path of branches is one storyline. The rules: MULTIPLY along a path ('this AND then that'), ADD between whole paths ('this storyline OR that one'). Without replacement, the second stage's fractions change — one item gone from the top AND the bottom.

Why it works

Multiplying works because the second event happens in the world the first created: 'red then red' means red (5/8 of the time), and then red again in what's left (4/7 of THOSE times) — a fraction of a fraction. Adding works because different complete paths can't both happen — they're mutually exclusive storylines.

When a question is asking for it

Any two-stage random process. 'Both/and' → one path, multiplied. 'Different colours', 'exactly one' → several paths, each multiplied, then added. 'AT LEAST one' → the long way is many paths; the short way is 1 − P(none), because 'none' is a single path.

How to hold it in your head

Every complete path is a possible story of the experiment, and its probability is the product of its steps. The question then just asks: which stories count? Multiply within a story; add the stories that qualify.

2 · The mistakes that cost marks

  • Adding along a branch instead of multiplying (or getting a 'probability' bigger than 1 — the alarm bell for it).
  • Keeping the same denominators without replacement. After taking a red, both the reds AND the total drop by one.
  • Counting only one of the ordered paths for 'different colours': red-then-blue AND blue-then-red both qualify.
  • Grinding out 'at least one' the long way and dropping a path. 1 − P(none) is shorter and safer.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

A box has 6 milk and 4 dark chocolates. Two are eaten, one after the other. Find P(both dark).

  1. First: P(dark) = 4/10.
  2. Second, given a dark one has been eaten: 3 dark left in a box of 9, so 3/9.
  3. Multiply along the path: 4/10 × 3/9 = 12/90 = 2/15.
  4. (An outcome reached by more than one path — 'one of each', 'both the same' — is the SUM of its paths, each multiplied out like this.)

Answer: 2/15

A basketball player scores each free throw with probability 0.7. She takes two. Find P(at least one score).

  1. 'At least one' has three storylines — but 'none' has just one.
  2. P(none) = 0.3 × 0.3 = 0.09.
  3. P(at least one) = 1 − 0.09 = 0.91.

Answer: 0.91

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — The probability of rain on Saturday is 0.3 and, independently, the probability of rain on Sunday is 0.4. Work out the probability it rains on BOTH days.
  2. Grade 7 — A bag contains 5 red and 3 blue counters. Two counters are taken at random WITHOUT replacement. Work out the probability both are red. Give your answer as a fraction in simplest form.
  3. Grade 8 — From the same bag (5 red, 3 blue, two taken without replacement), work out the probability the two counters are DIFFERENT colours. Give your answer as a fraction in simplest form.
  4. Grade 9 — A netball player scores each shot with the same probability, independently of other shots. The probability that she misses BOTH of two shots is 0.16. She now takes three shots. Work out the probability she scores AT LEAST one of the three.
  5. 9+ — A box contains 7 blue pens and 5 black pens. Two pens are taken at random without replacement. Work out the probability that the two pens are different colours. Give your answer as a fraction in simplest form.

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on Combined Events & Tree Diagrams, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Combined Events & Tree Diagrams works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue Combined Events & Tree Diagrams against Christiaan Huygens — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 9 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

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