Mathematics (Core)Exam code: 0580

Set Notation and Probability Diagrams

Probability

PractiseFlashcards

Building one

  • A tree diagram sets out two or more stages, with a branch for each outcome at each stage
  • The syllabus states the convention these papers follow: outcomes at the ends of the branches, probabilities beside them
  • Each pair or group of branches from one point must add to 1, which fills in any missing probability
  • When a question prints a tree diagram, it asks you to complete it first, and that alone carries marks
The layout of a two-stage tree diagram: the first experiment branches to A and not A with P(A) and P(not A) beside the branches, each of those is followed by a pair of branches to B and not B with P(B) and P(not B) beside them, and the probability of each of the four routes is written at its end as P(A and B), P(A and not B), P(not A and B) and P(not A and not B)

Reading one

  • Follow one complete route across the diagram and multiply the probabilities you pass, which gives that path its own probability
  • Add between separate paths when more than one outcome satisfies the question
A tree diagram for two sets of traffic lights: the first branches green with probability 5 over 7 and red with 2 over 7, the second green with 8 over 9 and red with 1 over 9, and each of the four routes is multiplied along to give 40, 5, 16 and 2 out of 63
Source: Tree diagrams by Save My Exams
  • The probabilities of all the complete paths add to 1
  • In these papers the first stage never changes the second, so each second pair of branches carries the same probabilities whichever first branch it follows: the first pair again when one spinner or bag is used twice, or a new pair for a different event, like the second set of traffic lights above

Exam tip

Multiply the values on your tree

The probability of a route is marked from your own tree: writing the multiplication with the values on your branches, such as 3/8 × 3/8 below, earns the method mark even if one of them is wrong. So always write that line before the answer. It counts only while each value is a probability between 0 and 1, so a branch value such as 8/3 is a slip to fix first.

Worked example

Two spins on a tree diagram

A spinner has 3 red sections and 5 blue sections, all the same size. It is spun twice. Find the probability that both spins are red, and the probability that exactly one spin is red.

Solution:

  • There are 8 sections, so each spin is red with probability 3/8 and blue with probability 5/8
  • The spinner is unchanged between spins, so the second pair of branches repeats the first
A completed tree diagram with the headings First spin and Second spin. The first pair of branches is Red 3/8 and Blue 5/8, and after each of them the second pair is again Red 3/8 and Blue 5/8. No route products or answers are shown.
  • Both red is a single route, so multiply along it: P(red, red) = 3/8 × 3/8 = 9/64
  • Exactly one red happens two ways: red then blue, or blue then red
  • P(red, blue) = 3/8 × 5/8 = 15/64, and P(blue, red) = 5/8 × 3/8 = 15/64
  • Add the separate routes: 15/64 + 15/64 = 30/64 = 15/32
  • Check: the four routes come to 9/64 + 15/64 + 15/64 + 25/64 = 1

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