Listing Outcomes and Sample Space Diagrams
A sample space is the complete list of every possible outcome of one or more events; listing outcomes systematically, in a clear order without missing any or repeating any, makes sure none are left out when working out a probability. A sample space diagram is a table, usually a grid, that organises all the possible outcomes of two combined events, such as rolling two dice or flipping a coin and rolling a dice, so that a probability can be read off directly by counting cells.
Before you start
Make sure you're comfortable with these topics first:
Method
- When listing the outcomes of a single event, work through them in a logical order (such as numerical or alphabetical) so none are missed or written twice.
- When listing the outcomes of two combined events, use ordered pairs, such as (Head, 3), keeping the events in the same order throughout.
- For two combined events, a sample space diagram is a grid with one event's outcomes along the top and the other event's outcomes down the side.
- Fill each cell of the grid with the combined outcome (or its value, such as a total) for that row and column.
- Count the total number of cells in the grid to find the total number of possible outcomes.
- To find a probability from the diagram, count the cells that match the event asked about, and divide by the total number of cells.
Worked example
Two fair 4-sided dice, each numbered 1 to 4, are rolled together and their scores are added. (a) Draw a sample space diagram to show all the possible totals. (b) Find the probability that the total is exactly 5.
- Draw a 4 by 4 grid, with the first dice's scores (1, 2, 3, 4) along the top and the second dice's scores (1, 2, 3, 4) down the side.
- Fill each cell with the sum of its row and column value, giving totals from 1 + 1 = 2 up to 4 + 4 = 8.
- The completed grid has 16 cells in total, so there are 16 equally likely outcomes.
- Count the cells showing a total of 5: (1,4), (2,3), (3,2) and (4,1), which is 4 cells.
- P(total is 5) = 4/16 = 1/4.
Practice questions
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Q1List all the possible outcomes when a coin is flipped and a fair 4-sided dice (numbered 1 to 4) is rolled together.Show answer
Answer: (H,1), (H,2), (H,3), (H,4), (T,1), (T,2), (T,3), (T,4) - 8 outcomes in total
Q2A fair 3-sided spinner is numbered 1, 2, 3 and a fair coin is flipped. List all the possible outcomes as ordered pairs.Show answer
Answer: (1,H), (1,T), (2,H), (2,T), (3,H), (3,T) - 6 outcomes in total
Q3Two fair coins are flipped together. List all the possible outcomes.Show answer
Answer: (H,H), (H,T), (T,H), (T,T) - 4 outcomes in total
Q4Two fair standard six-sided dice are rolled together and their scores are multiplied. How many cells would the sample space diagram for this situation have?Show answer
Answer: 36 (a 6 by 6 grid)
Q5Using a sample space diagram for two fair six-sided dice added together, find P(the total is exactly 7).Show answer
Answer: 1/6 (6/36; the pairs are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1))
Q6Using a sample space diagram for two fair six-sided dice added together, find P(the total is at least 10).Show answer
Answer: 1/6 (6/36; the pairs are (4,6), (5,5), (6,4), (5,6), (6,5), (6,6))
Q7A fair 5-sided spinner numbered 1 to 5 is spun twice and the two scores are added together. Find P(the total is an even number).Show answer
Answer: 13/25 (there are 25 equally likely outcomes in total; the total is even when both scores are odd, 3 x 3 = 9 outcomes, or both are even, 2 x 2 = 4 outcomes, giving 9 + 4 = 13 even totals out of 25)
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
A fair 4-sided dice (numbered 1 to 4) and a fair 6-sided dice (numbered 1 to 6) are rolled together and their scores are added. (a) State the total number of outcomes in the sample space. (b) Find the probability that the total is exactly 6.
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A fair coin is flipped and a fair 6-sided dice is rolled together. (a) List all 12 possible outcomes. (b) Find the probability of getting a head and a number greater than 4.
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Two fair six-sided dice are rolled together and the difference between the two scores (the larger score minus the smaller score) is recorded. (a) Show that there are 6 ways of getting a difference of 0. (b) Find the probability that the difference is exactly 3.
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Free printable worksheet
Want more practice on paper? Download the listing outcomes and sample space diagrams worksheet pack - 7 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 23 of KS3 Maths Workbook 2, the whole course as one free printable PDF.
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