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13.2 Two Events

GCSE Maths Foundation · Edexcel · Chapter 13: Probability

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: listing outcomes, sample space diagrams and two-way tables. 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

  • List all the outcomes of two events in a systematic way
  • Draw and use a sample space diagram for two events
  • Complete a two-way table from given information
  • Work out probabilities from lists, sample space diagrams and two-way tables

Key words

  • Sample space: The set of all possible outcomes of an experiment.
  • Sample space diagram: A grid that shows every outcome of two events, with one event along the top and the other down the side.
  • Two-way table: A table that sorts data by two things at once, for example age group and favourite sport.
  • Systematic: Done in a fixed order so that no outcome is missed or repeated.

Listing outcomes systematically

When two things happen, such as flipping a coin and spinning a spinner, list every pair. Keep the first result fixed and run through every second result, then change the first.

Worked example: a coin and a three-sided spinner

A fair coin is flipped and a fair spinner numbered 1, 2, 3 is spun.
Heads first: H1, H2, H3. Then tails: T1, T2, T3. That is 6 equally likely outcomes.
P(heads and an odd number) = P(H1 or H3) = 2/6 = 1/3.

Answer: 6 outcomes; P(heads and odd) = 1/3

Key fact: Number of outcomes for two events = (outcomes for the first) × (outcomes for the second). Here 2 × 3 = 6.

Sample space diagrams

When there are lots of outcomes, a grid is quicker and safer than a list. Each cell is one equally likely outcome.

  • 1; 1 2; 2 3; 3 4; 4 5
  • 2; 1 3; 2 4; 3 5; 4 6
  • 3; 1 4; 2 5; 3 6; 4 7
  • 4; 1 5; 2 6; 3 7; 4 8

Worked example: adding two spinners

There are 4 × 4 = 16 cells. A total of 5 appears 4 times (1+4, 2+3, 3+2, 4+1).
P(total of 5) = 4/16 = 1/4.
Totals of 7 or more: 7, 7, 8 – that is 3 cells, so P(total ≥ 7) = 3/16.

Answer: P(5) = 1/4, P(at least 7) = 3/16

The same idea works for any rule: add, subtract (difference), or multiply the two scores.

Completing two-way tables

Every row and every column adds up to its total. Fill in the gaps one at a time, using any row or column with only one blank.

Worked example: visitors to a museum

120 people visit a museum. 70 are adults. 45 people visit in the morning, and 28 of them are adults.
Adults in the afternoon = 70 − 28 = 42. Children = 120 − 70 = 50.
Children in the morning = 45 − 28 = 17. Children in the afternoon = 50 − 17 = 33.

Answer: see the table (drawn in the printable lesson notes)

  • Adults; Morning 28; Afternoon 42; Total 70
  • Children; Morning 17; Afternoon 33; Total 50
  • Total; Morning 45; Afternoon 75; Total 120

Key fact: Check: the grand total in the corner is the sum of the bottom row and also the sum of the right-hand column.

Probability from a two-way table

For “a person is picked at random from everyone”, divide by the grand total. If you are told the person is from one group, divide by that group’s total instead.

Worked example: using the museum table

(a) A visitor is picked at random. P(child in the afternoon) = 33/120 = 11/40.
(b) An adult is picked at random. P(visited in the morning) = 28/70 = 2/5.

Answer: (a) 11/40 (b) 2/5

Common mistakes

  • Missing outcomes by listing in a random order. Fix the first result, then change the second.
  • Thinking all totals are equally likely. With two dice, 7 can be made 6 ways but 12 only 1 way.
  • Counting HT and TH as the same outcome. With two coins there are 4 outcomes, not 3.
  • Dividing by the grand total when the question only picks from one row or column.

Revision questions

Try these, then check your answers below each one.

Question 1. A fair coin is flipped and a fair six-sided dice is rolled. How many different outcomes are there?

Answer: 12

Working: 2 × 6 = 12 outcomes.

Question 2. Two fair spinners are each numbered 1, 2, 3, 4. Both are spun and the scores are added. Work out P(total of 6). Give a fraction.

Answer: 3/16

Working: Totals of 6: 2+4, 3+3, 4+2. That is 3 of 16 outcomes, so 3/16.

Question 3. Two fair dice are rolled and the scores are added. Which total is most likely? (Choose from: 2, 6, 7, 12.)

Answer: 7

Working: 7 can be made 6 ways (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), more than any other total.

Related lessons

Continue in the full course

This lesson is section 13.2 (EGCMF1302) 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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