An arrangement of objects (such as numbers, pictures, algebraic expressions) in rows and columns is called an array. When listing outcomes a $2$2-dimensional array can be useful for organising and displaying all the possibilities. For example, the possible outcomes when rolling two die are:
$\left(1,1\right),\left(1,2\right),\left(1,3\right),\left(1,4\right),\left(1,5\right),\left(1,6\right),\left(2,1\right),\left(2,2\right),\left(2,3\right),\left(2,4\right),\left(2,5\right),\left(2,6\right),\left(3,1\right),\left(3,2\right),\left(3,3\right),\left(3,4\right),\left(3,5\right),\left(3,6\right),\left(4,1\right),\left(4,2\right),\left(4,3\right),\left(4,4\right),\left(4,5\right),\left(4,6\right),\left(5,1\right),\left(5,2\right),\left(5,3\right),\left(5,4\right),\left(5,5\right),\left(5,6\right),\left(6,1\right),\left(6,2\right),\left(6,3\right),\left(6,4\right),\left(6,5\right),\left(6,6\right)$(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),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)
While possible to list all outcomes it is not a convenient form to count or find outcomes with particular characteristics. We could instead create an array to display the outcomes in an organised form, this will also help to not miss any outcomes in the list. For our list above we could create the following array to display the outcomes:
| Die 1 | |||||||
|---|---|---|---|---|---|---|---|
| $1$1 | $2$2 | $3$3 | $4$4 | $5$5 | $6$6 | ||
|
Die 2 |
$1$1 | $1,1$1,1 | $1,2$1,2 | $1,3$1,3 | $1,4$1,4 | $1,5$1,5 | $1,6$1,6 |
| $2$2 | $2,1$2,1 | $2,2$2,2 | $2,3$2,3 | $2,4$2,4 | $2,5$2,5 | $2,6$2,6 | |
| $3$3 | $3,1$3,1 | $3,2$3,2 | $3,3$3,3 | $3,4$3,4 | $3,5$3,5 | $3,6$3,6 | |
| $4$4 | $4,1$4,1 | $4,2$4,2 | $4,3$4,3 | $4,4$4,4 | $4,5$4,5 | $4,6$4,6 | |
| $5$5 | $5,1$5,1 | $5,2$5,2 | $5,3$5,3 | $5,4$5,4 | $5,5$5,5 | $5,6$5,6 | |
| $6$6 | $6,1$6,1 | $6,2$6,2 | $6,3$6,3 | $6,4$6,4 | $6,5$6,5 | $6,6$6,6 | |
This is particularly useful for experiments with two independent events such as tossing two coins, rolling two dice or selecting two marbles from a bag with replacement.
Worked example
A player is rolling two dice and calculating the sum of the of the two numbers shown.
(a) Create an array displaying all outcomes of the event.
Think: Create a table with the results of one die shown horizontally and the other die vertically. Find the sum of each corresponding pair and record the sum.
| Die 1 | |||||||
|---|---|---|---|---|---|---|---|
| $1$1 | $2$2 | $3$3 | $4$4 | $5$5 | $6$6 | ||
|
Die 2 |
$1$1 | $\editable{1+1}$1+1 | $\editable{1+2}$1+2 | $\editable{1+3}$1+3 | $\editable{1+4}$1+4 | $\editable{1+5}$1+5 | $\editable{1+6}$1+6 |
| $2$2 | $\editable{2+1}$2+1 | $\editable{2+2}$2+2 | $\editable{2+3}$2+3 | $\editable{2+4}$2+4 | $\editable{2+5}$2+5 | $\editable{2+6}$2+6 | |
| $3$3 | $\dots$… | ||||||
| $4$4 | |||||||
| $5$5 | |||||||
| $6$6 | |||||||
Do:
| Die 1 | |||||||
|---|---|---|---|---|---|---|---|
| $1$1 | $2$2 | $3$3 | $4$4 | $5$5 | $6$6 | ||
|
Die 2 |
$1$1 | $2$2 | $3$3 | $4$4 | $5$5 | $6$6 | $7$7 |
| $2$2 | $3$3 | $4$4 | $5$5 | $6$6 | $7$7 | $8$8 | |
| $3$3 | $4$4 | $5$5 | $6$6 | $7$7 | $8$8 | $9$9 | |
| $4$4 | $5$5 | $6$6 | $7$7 | $8$8 | $9$9 | $10$10 | |
| $5$5 | $6$6 | $7$7 | $8$8 | $9$9 | $10$10 | $11$11 | |
| $6$6 | $7$7 | $8$8 | $9$9 | $10$10 | $11$11 | $12$12 | |
(b) State the total number of possible outcomes.
Think: The grid of outcomes has $6$6 rows and $6$6 columns.
Do: The total number of outcomes from rolling two dice is $6\times6=36$6×6=36.
Reflect: Not all the outcomes are different so our sample space is $S=\left\{2,3,4,5,6,7,8,9,10,11,12\right\}$S={2,3,4,5,6,7,8,9,10,11,12}, however they are not all equally likely.
(c) What is the most common outcome and what is its probability?
Think: Which number appears most often in the table and how many times does it appear out of the $36$36 outcomes?
Do: Seven is the most common result from rolling two dice and finding the sum. This occurs $6$6 times and hence:
| $P\left(\text{Seven}\right)$P(Seven) | $=$= | $\frac{6}{36}$636 |
| $=$= | $\frac{1}{6}$16 |
(d) What is the probability of a sum less than $5$5?
Think: Possible results less than five are $2$2, $3$3 and $4$4, how many of these are there?
Do:
| $P\left(\text{Result}<5\right)$P(Result<5) | $=$= | $\frac{6}{36}$636 |
| $=$= | $\frac{1}{6}$16 |
(e) What is the probability of a sum greater than $6$6 and even?
Think: Possible results greater than six that are also even are $8$8, $10$10 and $12$12, how many of these are there?
Do:
| $P\left(\text{Result}>6\text{ and even}\right)$P(Result>6 and even) | $=$= | $\frac{9}{36}$936 |
| $=$= | $\frac{1}{4}$14 |
Practice questions
QUESTION 1
Two dice are rolled, and the combination of numbers rolled on the dice is recorded.
Complete the table of outcomes such that each entry in the table consists of the number rolled on Die 1 followed by the number rolled by Die 2.
Die 2 1 2 3 4 5 6 1 11 12 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ 2 21 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Die 1 3 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ 4 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ 5 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ 6 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Find $P$P$($(rolling a $1$1 and a $4$4$)$).
Find $P$P$($(rolling a $1$1 and then a $4$4$)$).
Find $P$P$($(a difference of $4$4 between the two numbers$)$).
Find $P$P$($(a product of $12$12$)$).
Find $P$P$($( the difference between the two numbers is no more than $2$2$)$).
The numbers appearing on the uppermost faces are added. Which of the following are true?
A sum greater than $7$7 and a sum less than $7$7 are equally likely.
AA sum greater than $7$7 is more likely than a sum less than $7$7.
BA sum of $5$5 or $9$9 is more likely than a sum of $4$4 or $10$10.
CAn even sum is more likely than an odd sum.
D
Question 2
The following two spinners are spun and the sum of their respective spins are recorded.
Complete the following table to represent all possible combinations.
Spinner $2$2 $3$3 $4$4 $7$7 $9$9 $\editable{}$ $\editable{}$ $9$9 $\editable{}$ $\editable{}$ $13$13 $12$12 $\editable{}$ $\editable{}$ $\editable{}$ State the total number of possible outcomes.
What is the probability that the first spinner lands on an even number and the sum is even?
What is the probability that the first spinner lands on a prime number and the sum is odd?
What is the probability that the sum is a multiple of $4$4?