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13.5 Tree Diagrams

GCSE Maths Foundation · Edexcel · Chapter 13: Probability

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: frequency trees, independent events and probability tree diagrams. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Complete and use frequency trees
  • Use the AND rule P(A and B) = P(A) × P(B) for independent events
  • Draw probability tree diagrams for two events
  • Find probabilities of combined outcomes by multiplying along branches and adding the results

Key words

  • Frequency tree: A diagram that splits a total number of people or things into groups, and then into smaller groups.
  • Independent events: Events where the outcome of one does not change the probability of the other, like two flips of a coin.
  • Tree diagram: A diagram with a branch for each outcome, labelled with its probability.
  • Combined event: An outcome made of two or more events, such as “late on Monday and on Tuesday”.

Frequency trees

A frequency tree starts with a total and splits it twice. The numbers on each pair of branches add up to the number they came from.

Worked example: using the frequency tree

120 travel by train, so 200 − 120 = 80 travel by bus. Of the bus users, 56 are on time, so 80 − 56 = 24 are late.
A commuter is picked at random. P(bus and late) = 24/200 = 3/25.
P(late) = (18 + 24)/200 = 42/200 = 21/100.

Answer: 3/25 and 21/100

Independent events: the AND rule

If two events are independent, multiply their probabilities to find the probability that both happen.

Key fact: P(A and B) = P(A) × P(B) when A and B are independent

Worked example: a coin and a dice

A fair coin is flipped and a fair dice is rolled. Find P(heads and a 6).
P(heads) = 1/2, P(6) = 1/6. 1/2 × 1/6 = 1/12.

Answer: 1/12

Check with a list: there are 2 × 6 = 12 equally likely outcomes, and only H6 works.

Drawing a probability tree diagram

Draw one set of branches for the first event and a set for the second event from the end of every first branch. Write each probability on its branch. Each pair of branches from one point adds up to 1.

Using a tree diagram

Key fact: Multiply along the branches for “and”. Add the results of different paths for “or”.

Worked example: the late bus

P(late both days) = 0.2 × 0.2 = 0.04.
P(late on exactly one day) = P(L, not L) + P(not L, L) = 0.16 + 0.16 = 0.32.
P(late on at least one day) = 1 − P(not late either day) = 1 − 0.64 = 0.36.

Answer: 0.04, 0.32, 0.36

Check: the four end probabilities add up to 1: 0.04 + 0.16 + 0.16 + 0.64 = 1.

Common mistakes

  • Adding along the branches instead of multiplying. P(late both days) = 0.2 × 0.2, not 0.2 + 0.2.
  • Forgetting the second path for “exactly one”: late then not late, or not late then late.
  • Branch pairs that do not add up to 1.
  • In a frequency tree, putting a total for the whole group on a branch that should only count part of it.

Revision questions

Try these, then check your answers below each one.

Question 1. A fair coin is flipped and a fair dice is rolled. Work out P(tails and an even number) as a decimal.

Answer: 0.25

Working: 1/2 × 1/2 = 0.25.

Question 2. Events A and B are independent. P(A) = 0.6 and P(B) = 0.3. Work out P(A and B).

Answer: 0.18

Working: 0.6 × 0.3 = 0.18.

Question 3. P(rain on Saturday) = 0.3 and P(rain on Sunday) = 0.3, independently. Jon says P(rain on both days) = 0.6. Is Jon correct?

Answer: No

Working: No. Multiply: 0.3 × 0.3 = 0.09.

Related lessons

Continue in the full course

This lesson is section 13.5 (EGCMF1305) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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