 # Use Systematic Listing Strategies

In this worksheet, students will employ systematic listing strategies to find all possible outcomes of linked events, which can then be used to calculate the probability of specific outcomes. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Number, Probability

Curriculum subtopic:   Structure and Calculation, Combined Events and Probability Diagrams

Difficulty level:   ### QUESTION 1 of 10

You may be given a question where there are multiple different options to choose from and it's not immediately obvious which option to choose.

If you get a question like this, you need to use a systematic listing strategy to write out all possible combinations.

e.g. A shop sells 'Roast Beef', 'Ready Salted' and 'Salt & Vinegar' flavoured crisps, plus lemonade and cola as drinks options.

If you wanted to buy one bag of crisps and a drink, how many options do you have?

To solve this, we need to look at one of the options first and see how many of the other options could be paired with this.

So: 'Roast Beef' crisps could be bought with either cola or lemonade  >  (B,C   B,L)

Next: 'Ready Salted' crisps could be bought with either cola or lemonade  >  (P,C   P,L)

Lastly: 'Salt & Vinegar' crisps could be bought with cola or lemonade  >  (S,C   S,L)

From this, we can see that there are 6 different options in total.

These questions are frequently paired with probability.

e.g. A shop sells 'Roast Beef', 'Ready Salted' and 'Salt & Vinegar' flavoured crisps, plus lemonade and cola as drinks options.

If someone randomly bought one bag of crisps and a drink, what is the probability they bought 'Roast Beef' crisps with a lemonade?

We know that there are 6 options in total and only one of these is the combination described in the question.

The probability of this choice will therefore be 1/6.

In this activity, you will employ systematic listing strategies to find all possible outcomes of linked events, which can then be used to calculate the probability of specific outcomes.

What do you need to make a list?

That's right - a pen and paper!

Complete the blank below to define the concept of systematic listing

If we flip two coins at the same time, how many different outcomes are possible?

In a deli, customers can choose to have either plain, Italian, cheesy or rye bread for sandwiches.

They can fill this with either ham, cheese or beef.

How many different sandwiches can customers choose from overall?

7

12

9

To follow on from the previous question.

In a deli, customers can choose to have either plain, Italian, cheesy or rye bread for sandwiches.

They can fill this with either ham, cheese or beef.

What is the probability that a customer chooses beef on rye bread?

Write your answer as a fraction of the form a/b, with no spaces and using the / key as your fraction bar.

Match the situations below to their total number of possible combinations.

## Column B

5 options for A, 4 options for B
4
3 options for A, 6 options for B
36
2 options for A, 2 options for B
18
6 options for A, 6 options for B
20

Emmanuel has three scrabble tiles with A, B and C on them.

Without looking, he picks one up and then another.

How many different combinations could he achieve?

Emma chose two vowels at random from the alphabet.

How many different outcomes are possible?

5

20

25

This time, Emma chooses two vowels again.

However, she decides that now she cannot pick the same vowel twice.

How many different outcomes are possible?

5

20

25

Saranya picks a number between 1 and 9, then she does this again.

How many different outcomes are possible?

Saranya picks a number between 1 and 9, then she does this again.

What is the probability that she has chosen the numbers 3 and 7?

Write your answer as a fraction of the form a/b, with no spaces and using the / key as your fraction bar.

• Question 1

Complete the blank below to define the concept of systematic listing

EDDIE SAYS
We use systematic listing to write out all the scenarios that could possibly take place. We call these 'the combinations of outcomes'.
• Question 2

If we flip two coins at the same time, how many different outcomes are possible?

4
EDDIE SAYS
Let's think about this logically. If we get heads on one coin, we can then get either heads or tails on the other: HH HT If we get tails on one coin, we can then get either heads or tails on the other: TH TT So in total there are 4 possible outcomes: HH HT TH TT Even though HT and TH technically have the same composition (one heads and one tails), they are classed as different events as the outcomes are achieved in a different order.
• Question 3

In a deli, customers can choose to have either plain, Italian, cheesy or rye bread for sandwiches.

They can fill this with either ham, cheese or beef.

How many different sandwiches can customers choose from overall?

12
EDDIE SAYS
For each type of bread, 3 different fillings are possible. 4 x 3 = 12 If we list these outcomes they are: Plain + ham, plain + cheese, plain + beef Italian + ham, Italian + cheese, Italian + beef Cheesy + ham, cheesy + cheese, cheesy + beef Rye + ham, rye + cheese, rye + beef Is this beginning to make more sense? Let's practise some more.
• Question 4

To follow on from the previous question.

In a deli, customers can choose to have either plain, Italian, cheesy or rye bread for sandwiches.

They can fill this with either ham, cheese or beef.

What is the probability that a customer chooses beef on rye bread?

Write your answer as a fraction of the form a/b, with no spaces and using the / key as your fraction bar.

1/12
EDDIE SAYS
As we found in the previous question, there are 12 different possible options. Only one of these outcomes describes the scenario in the question (beef on rye). This means that the probability of this choice is 1/12.
• Question 5

Match the situations below to their total number of possible combinations.

## Column B

5 options for A, 4 options for B
20
3 options for A, 6 options for B
18
2 options for A, 2 options for B
4
6 options for A, 6 options for B
36
EDDIE SAYS
Let's look at one of these as an example: 5 options for A, 4 options for B For each of the 5 options for A, there are 4 for B. So there are 4 + 4 + 4 + 4 + 4 combinations possible in total. A quick tricky to find out how many combinations there are in total is to multiply the number of options together. Can you see this in action here?
• Question 6

Emmanuel has three scrabble tiles with A, B and C on them.

Without looking, he picks one up and then another.

How many different combinations could he achieve?

6
EDDIE SAYS
This scenario is a bit different from the others we have explored, because after one tile has been picked, only 2 tiles remain for the second pick. So Emmanuel's options are: AB AC BA BC CA CB Which means there are 6 possible outcomes in total, or 3 x 2.
• Question 7

Emma chose two vowels at random from the alphabet.

How many different outcomes are possible?

25
EDDIE SAYS
In this question, it is possible for Emma to choose the same vowel twice so the totals stay the same in each case, even though she is picking twice. There are 5 vowels in the alphabet (a, e, i, o, u) so the total number of combinations can be found using: 5 x 5 = 25
• Question 8

This time, Emma chooses two vowels again.

However, she decides that now she cannot pick the same vowel twice.

How many different outcomes are possible?

20
EDDIE SAYS
Emma has 5 options for her first pick but only 4 for her second, as she is not allowed to pick the same vowel twice with her new rules. Therefore, the total amount of possible outcomes can be calculated using the sum: 5 x 4 = 20
• Question 9

Saranya picks a number between 1 and 9, then she does this again.

How many different outcomes are possible?

81
EDDIE SAYS
There are 9 options for the first number Saranya picks and 9 for the second. This gives her 81 possible combinations in total (9 x 9). Just one challenge remains to complete this activity!
• Question 10

Saranya picks a number between 1 and 9, then she does this again.

What is the probability that she has chosen the numbers 3 and 7?

Write your answer as a fraction of the form a/b, with no spaces and using the / key as your fraction bar.

2/81
EDDIE SAYS
There are 9 options for the first number Saranya picks and 9 for the second. This gives her 81 possible combinations in total (9 x 9). The question doesn't say that the numbers 3 and 7 have to be chosen in a particular order. So 3, 7 and 7, 3 are both valid combinations. This means that 2 possible combinations out of 81 satisfy the criteria. Great work, you've completed another activity!
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