a Find how the outcomes result in a sum of 8 one by one.
B
b Find how the outcomes result in an odd sum one by one.
A
a 5 outcomes
B
b 18 outcomes
Practice makes perfect
a We are told that Cody and Monette are playing a board game in which they roll two dice per turn. We want to find the number of outcomes that result in a sum of 8. Let's do it.
Number in the First Dice
Number in the Second Dice
Sum
2
6
2+ 6=8
3
5
3+ 5=8
4
4
4+ 4=8
5
3
5+ 3=8
6
2
6+ 2=8
There are 5 outcomes.
b We want to find the number of outcomes in one turn that result in an odd sum. To obtain an odd sum the number on one dice needs to be odd, whereas the number on the other dice needs to be even. Let's list the possible outcomes.
Number on the First Dice
Number on the Second Dice
Sum
Number on the First Dice
Number on the Second Dice
Sum
1
2
1+ 2=3
4
1
4+ 1=5
1
4
1+ 4=5
4
3
4+ 3=7
1
6
1+ 6=7
4
5
4+ 5=9
2
1
2+ 1=3
5
2
5+ 2=7
2
3
2+ 3=5
5
4
5+ 4=9
2
5
2+ 5=7
5
6
5+ 6=11
3
2
3+ 2=5
6
1
6+ 1=7
3
4
3+ 4=7
6
3
6+ 3=9
3
6
3+ 6=9
6
5
6+ 5=11
Therefore, there are 18 outcomes of the desired situation.
Alternative Solution
Using the Fundamental Counting Principle
We can also find the number of outcomes by using the Fundamental Counting Principle. We know that to have an odd sum the number on one dice has to be odd, whereas the number on the other dice has to be even. We have two cases.
The number on the first dice is odd.
The number on the second dice is odd.
Let's first find the number of outcomes of the first case.
Case I
We will identify the number of possible outcomes of each event.
Possible Outcomes
Type of the Outcomes
Number of Outcomes
First Dice
1,3,5
Odd
3
Second Dice
2,4,6
Even
3
We will find the number of possible outcomes when the number on the first dice is odd and the number on the second dice is even.
3 * 3=9
There are 9 outcomes.
Case II
We will apply the same steps for Case II. This time, the number on the first dice will be even.
Possible Outcomes
Type of the Outcomes
Number of Outcomes
First Dice
2,4,6
Even
3
Second Dice
1,3,5
Odd
3
We will use the principle to find the number of possible outcomes when the number on the first dice is even and the number on the second dice is odd.
3 * 3=9
This case also has 9 outcomes. We will find the total number of outcomes of the desired situation by adding up the outcomes that we found.
9+9=18
