Sec 4 E-Math · Probability

Possibility diagrams questions with answers

Practice questions on possibility diagrams, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

Two fair dice are thrown. Using a possibility diagram, find the probability that the product of the two numbers is $4$ .

×	1	2	3	4	5	6
1
2
3
4
5
6

First die down the side, second die across the top

Step 1 of 3

Fill in the possibility diagram

There are

6 \times 6 = 36

equally likely outcomes.

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Questions

Question 1Two dice: an exact total
Two fair dice are thrown. Using a possibility diagram, find the probability that the sum of the two numbers is $2$ .
+ 1 2 3 4 5 6
1
2
3
4
5
6
First die down the side, second die across the top
Show the worked solution
1. Step 1: Fill in the possibility diagramThere are $6 \times 6 = 36$ equally likely outcomes.
2. Step 2: Count the outcomes that work $(1, 1)$ That is $1$ outcomes.
3. Step 3: Divide by the total $P = \dfrac{1}{36}$
Question 2Two dice: an exact total
Two fair dice are thrown. Using a possibility diagram, find the probability that the sum of the two numbers is $3$ .
+ 1 2 3 4 5 6
1
2
3
4
5
6
First die down the side, second die across the top
Show the worked solution
1. Step 1: Fill in the possibility diagramThere are $6 \times 6 = 36$ equally likely outcomes.
2. Step 2: Count the outcomes that work $(1, 2),\ (2, 1)$ That is $2$ outcomes.
3. Step 3: Divide by the total and simplify $\mathstrut P = \dfrac{2}{36}$ $P = \dfrac{1}{18}$
Question 3Two dice: a condition
Two fair dice are thrown. Using a possibility diagram, find the probability that the sum of the two numbers is less than $10$ .
+ 1 2 3 4 5 6
1
2
3
4
5
6
First die down the side, second die across the top
Show the worked solution
1. Step 1: Fill in the possibility diagramThere are $6 \times 6 = 36$ equally likely outcomes.
2. Step 2: Count the outcomes that work $(1, 1),\ (1, 2),\ (1, 3),\ (1, 4),\ (1, 5),\ (1, 6),\ (2, 1),\ (2, 2),\ (2, 3),\ (2, 4),\ (2, 5),\ (2, 6),\ (3, 1),\ (3, 2),\ (3, 3),\ (3, 4),\ (3, 5),\ (3, 6),\ (4, 1),\ (4, 2),\ (4, 3),\ (4, 4),\ (4, 5),\ (5, 1),\ (5, 2),\ (5, 3),\ (5, 4),\ (6, 1),\ (6, 2),\ (6, 3)$ That is $30$ outcomes.
3. Step 3: Divide by the total and simplify $\mathstrut P = \dfrac{30}{36}$ $P = \dfrac{5}{6}$
Question 4Two dice: a condition
Two fair dice are thrown. Using a possibility diagram, find the probability that the sum of the two numbers is greater than $8$ .
+ 1 2 3 4 5 6
1
2
3
4
5
6
First die down the side, second die across the top
Show the worked solution
1. Step 1: Fill in the possibility diagramThere are $6 \times 6 = 36$ equally likely outcomes.
2. Step 2: Count the outcomes that work $(3, 6),\ (4, 5),\ (4, 6),\ (5, 4),\ (5, 5),\ (5, 6),\ (6, 3),\ (6, 4),\ (6, 5),\ (6, 6)$ That is $10$ outcomes.
3. Step 3: Divide by the total and simplify $\mathstrut P = \dfrac{10}{36}$ $P = \dfrac{5}{18}$
Question 5Two spinners
Spinner A has the numbers $1, 2, 3$ and spinner B has $1, 3, 5, 7$ . Both are spun. Using a possibility diagram, find the probability that the product of the two numbers is greater than $6$ .
× 1 3 5 7
1
2
3
Spinner A down the side, spinner B across the top
Show the worked solution
1. Step 1: Fill in the possibility diagramThere are $3 \times 4 = 12$ equally likely outcomes.
2. Step 2: Count the outcomes that work $(1, 7),\ (2, 5),\ (2, 7),\ (3, 3),\ (3, 5),\ (3, 7)$ That is $6$ outcomes.
3. Step 3: Divide by the total and simplify $\mathstrut P = \dfrac{6}{12}$ $P = \dfrac{1}{2}$
Question 6Two spinners
Spinner A has the numbers $1, 2, 3$ and spinner B has $1, 2, 3, 4$ . Both are spun. Using a possibility diagram, find the probability that the sum of the two numbers is greater than $5$ .
+ 1 2 3 4
1
2
3
Spinner A down the side, spinner B across the top
Show the worked solution
1. Step 1: Fill in the possibility diagramThere are $3 \times 4 = 12$ equally likely outcomes.
2. Step 2: Count the outcomes that work $(2, 4),\ (3, 3),\ (3, 4)$ That is $3$ outcomes.
3. Step 3: Divide by the total and simplify $\mathstrut P = \dfrac{3}{12}$ $P = \dfrac{1}{4}$

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Worked example

Two fair dice are thrown. Using a possibility diagram, find the probability that the product of the two numbers is

4

×	1	2	3	4	5	6
1
2
3
4
5
6

First die down the side, second die across the top

Step 1 of 3

Fill in the possibility diagram

There are

6 \times 6 = 36

equally likely outcomes.

Questions

Question 1Two dice: an exact total

Two fair dice are thrown. Using a possibility diagram, find the probability that the sum of the two numbers is $2$ .

+	1	2	3	4	5	6
1
2
3
4
5
6

First die down the side, second die across the top

Show the worked solution

Step 1: Fill in the possibility diagramThere are $6 \times 6 = 36$ equally likely outcomes.
Step 2: Count the outcomes that work $(1, 1)$ That is $1$ outcomes.
Step 3: Divide by the total $P = \dfrac{1}{36}$

Question 2Two dice: an exact total

Two fair dice are thrown. Using a possibility diagram, find the probability that the sum of the two numbers is $3$ .

+	1	2	3	4	5	6
1
2
3
4
5
6

First die down the side, second die across the top

Show the worked solution

Step 1: Fill in the possibility diagramThere are $6 \times 6 = 36$ equally likely outcomes.
Step 2: Count the outcomes that work $(1, 2),\ (2, 1)$ That is $2$ outcomes.
Step 3: Divide by the total and simplify $\mathstrut P = \dfrac{2}{36}$ $P = \dfrac{1}{18}$

Question 3Two dice: a condition

Two fair dice are thrown. Using a possibility diagram, find the probability that the sum of the two numbers is less than $10$ .

+	1	2	3	4	5	6
1
2
3
4
5
6

First die down the side, second die across the top

Show the worked solution

Step 1: Fill in the possibility diagramThere are $6 \times 6 = 36$ equally likely outcomes.
Step 2: Count the outcomes that work $(1, 1),\ (1, 2),\ (1, 3),\ (1, 4),\ (1, 5),\ (1, 6),\ (2, 1),\ (2, 2),\ (2, 3),\ (2, 4),\ (2, 5),\ (2, 6),\ (3, 1),\ (3, 2),\ (3, 3),\ (3, 4),\ (3, 5),\ (3, 6),\ (4, 1),\ (4, 2),\ (4, 3),\ (4, 4),\ (4, 5),\ (5, 1),\ (5, 2),\ (5, 3),\ (5, 4),\ (6, 1),\ (6, 2),\ (6, 3)$ That is $30$ outcomes.
Step 3: Divide by the total and simplify $\mathstrut P = \dfrac{30}{36}$ $P = \dfrac{5}{6}$

Question 4Two dice: a condition

Two fair dice are thrown. Using a possibility diagram, find the probability that the sum of the two numbers is greater than $8$ .

+	1	2	3	4	5	6
1
2
3
4
5
6

First die down the side, second die across the top

Show the worked solution

Step 1: Fill in the possibility diagramThere are $6 \times 6 = 36$ equally likely outcomes.
Step 2: Count the outcomes that work $(3, 6),\ (4, 5),\ (4, 6),\ (5, 4),\ (5, 5),\ (5, 6),\ (6, 3),\ (6, 4),\ (6, 5),\ (6, 6)$ That is $10$ outcomes.
Step 3: Divide by the total and simplify $\mathstrut P = \dfrac{10}{36}$ $P = \dfrac{5}{18}$

Question 5Two spinners

Spinner A has the numbers $1, 2, 3$ and spinner B has $1, 3, 5, 7$ . Both are spun. Using a possibility diagram, find the probability that the product of the two numbers is greater than $6$ .

×	1	3	5	7
1
2
3

Spinner A down the side, spinner B across the top

Show the worked solution

Step 1: Fill in the possibility diagramThere are $3 \times 4 = 12$ equally likely outcomes.
Step 2: Count the outcomes that work $(1, 7),\ (2, 5),\ (2, 7),\ (3, 3),\ (3, 5),\ (3, 7)$ That is $6$ outcomes.
Step 3: Divide by the total and simplify $\mathstrut P = \dfrac{6}{12}$ $P = \dfrac{1}{2}$

Question 6Two spinners

Spinner A has the numbers $1, 2, 3$ and spinner B has $1, 2, 3, 4$ . Both are spun. Using a possibility diagram, find the probability that the sum of the two numbers is greater than $5$ .

+	1	2	3	4
1
2
3

Spinner A down the side, spinner B across the top

Show the worked solution

Step 1: Fill in the possibility diagramThere are $3 \times 4 = 12$ equally likely outcomes.
Step 2: Count the outcomes that work $(2, 4),\ (3, 3),\ (3, 4)$ That is $3$ outcomes.
Step 3: Divide by the total and simplify $\mathstrut P = \dfrac{3}{12}$ $P = \dfrac{1}{4}$