 # Using Sample Space Diagrams

In this worksheet, students practise listing all the outcomes of events by using a sample space diagram. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR

Curriculum topic:   Probability

Curriculum subtopic:   Probability, Combined Events and Probability Diagrams

Difficulty level:   ### QUESTION 1 of 10

What is a sample space diagram.

They are used to show all the outcomes of two events where the events are equally likely.

For example, you could use a sample space diagram when you roll dice, flip coins etc...

What does it look like?

All sample space diagrams have the same form, the outcomes for each event go along the top and the side and the outcomes for the combined events go in the middle. How do we fill one of these in?

This is easiest if we use an example.

I roll a dice and add the numbers together, draw a sample space diagram to show all the outcomes.

Step 1: Set up the outcomes for the individual events.

We know that on each dice, we have the numbers 1 to 6. Step 2: Fill in the inside

Ill start with a random one, we are told that we add the numbers. If we roll two fours and add, we will get 8 We can now fill in the rest. Will they always have numbers in the middle?

It depends on the question, if we looked at, for example, rolling a dice and flipping a coin, we can't add them together (or do anything else) so we just write both the outcomes. Using a sample space diagram to find probabilities.

Example: I roll two dice and add them together, what is the chance my numbers add to a prime number?

We already know that the sample space diagram looks like the one above. If we want to find the probability that my result is a prime, we count all the ones that are prime. From this, we can count that we have 15 primes.

We have 36 numbers in the middle section (we don't count the outside numbers)

This gives a probability of 15/36 which can cancel to 5/12

A sample space diagram is used to systematically list the...

Two dice are thrown and their score added, what numbers should be at positions A,B,C, D and E? ## Column B

A
4
B
5
C
9
D
12
E
8

Two dice are thrown and their score added, Find the probability I get 7. Two dice are thrown and their score added, Find the probability I get a number greater than 8 5/18

5/12

A Dice and a coin are thrown together, match the position with the outcome. ## Column B

A
T5
B
1T
C
H3

A Dice and a coin are thrown together, Find the probability I get a Head and an Even. 3/12

1/4

A Dice and a coin are thrown together, Find the probability I get Tails and a number greater than 4 Two dice are thrown and their score are subtracted, what numbers should be at positions A,B,C, D and E? ## Column B

A
3
B
3
C
0
D
0
E
2

Two dice are thrown and their score added, Find the probability I have a difference of 1. Two dice are thrown and their score added, Find the probability I have a difference of more than 2. 1/6

1/3

• Question 1

A sample space diagram is used to systematically list the...

EDDIE SAYS
Sample space diagrams only have one situation where we use them. This is to list all the outcomes for two events.
• Question 2

Two dice are thrown and their score added, what numbers should be at positions A,B,C, D and E? ## Column B

A
4
B
5
C
8
D
9
E
12
EDDIE SAYS
Remember when you are completing these, we have to look at the questions carefully. This one asks you to add. All you need to be looking at for a specific square are the number at the top of the column and at the start of the row.
• Question 3

Two dice are thrown and their score added, Find the probability I get 7. 1/6
EDDIE SAYS
If we just count how many ways to get a 7, there are 6 options (1 and 6, 2 and 5 etc). There are 36 ways in total to get a result. This gives a probability of 6/36 which cancels down to,..
• Question 4

Two dice are thrown and their score added, Find the probability I get a number greater than 8 5/18
EDDIE SAYS
This one catches people all the time. If the questions says greater than 8, it doesn't include the number 8. This means only 9, 10, 11 and 12 will work for this question, these options happen 10 ways. There are 36 ways in total to get a result. This gives a probability of 10/36 which cancels down to...
• Question 5

A Dice and a coin are thrown together, match the position with the outcome. ## Column B

A
H3
B
T5
C
1T
EDDIE SAYS
When you have one like this, you can't add the outcomes or do anything else, so we have to list the outcomes. Does it matter that B is T5 and C is 1T? Does it matter which way round you put these? Not really, but I would do it the same for all of them instead of mixing and matching.
• Question 6

A Dice and a coin are thrown together, Find the probability I get a Head and an Even. 3/12
1/4
EDDIE SAYS
We can see from the table that there are 3 ways of getting heads and an even number (H2, H4, H6) and 12 outcomes in total. This gives a probability of 3/12 which cancel to 1/4 Did you notice I didn't ask you to cancel? Both answers would be accepted as correct.
• Question 7

A Dice and a coin are thrown together, Find the probability I get Tails and a number greater than 4 1/6
EDDIE SAYS
We can see from the table that there are 2 ways of getting a tail and a number greater than 4 (T5, T6) and 12 outcomes in total. This gives a probability of 2/12 which cancel to...
• Question 8

Two dice are thrown and their score are subtracted, what numbers should be at positions A,B,C, D and E? ## Column B

A
0
B
3
C
2
D
3
E
0
EDDIE SAYS
Remember when you are completing these, we have to look at the questions carefully. This one asks you to subtract. All you need to be looking at for a specific square are the number at the top of the column and at the start of the row.
• Question 9

Two dice are thrown and their score added, Find the probability I have a difference of 1. 5/18
EDDIE SAYS
If we just count how many ways to get a difference of 1, there are 10 options. There are 36 ways in total to get a result. This gives a probability of 10/36 which cancels down to,..
• Question 10

Two dice are thrown and their score added, Find the probability I have a difference of more than 2. 1/3
EDDIE SAYS
If we just count how many ways to get a difference of more than 2 there are 12 options. There are 36 ways in total to get a result. This gives a probability of 12/36 which cancels down to,..
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