Answer :
Let's solve the problem step-by-step:
1. Identify Total Outcomes: When rolling two dice, each die has 6 faces. Therefore, the total number of possible outcomes when rolling two dice is [tex]\( 6 \times 6 = 36 \)[/tex].
2. Outcomes with Sum Less Than 9:
We need to determine the number of outcomes where the sum of the numbers on the two dice is less than 9. These outcomes include:
- Sum 2: (1,1)
- Sum 3: (1,2), (2,1)
- Sum 4: (1,3), (2,2), (3,1)
- Sum 5: (1,4), (2,3), (3,2), (4,1)
- Sum 6: (1,5), (2,4), (3,3), (4,2), (5,1)
- Sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
- Sum 8: (2,6), (3,5), (4,4), (5,3), (6,2)
Counting these outcomes gives us:
- 1 outcome for sum 2
- 2 outcomes for sum 3
- 3 outcomes for sum 4
- 4 outcomes for sum 5
- 5 outcomes for sum 6
- 6 outcomes for sum 7
- 5 outcomes for sum 8
Adding these up, the total number of outcomes where the sum is less than 9 is [tex]\(1 + 2 + 3 + 4 + 5 + 6 + 5 = 26\)[/tex].
3. Outcomes with Doubles and Sum Less Than 9:
Now, we need to find how many of these outcomes where the sum is less than 9 are doubles:
- For sum 4: (2,2)
- For sum 6: (3,3)
- For sum 8: (4,4)
Each of these sums has 1 double outcome:
- Total outcomes with doubles = 3
4. Calculate the Probability:
The probability that doubles are rolled, given that the sum is less than 9, is the ratio of the number of double outcomes to the total number of outcomes where the sum is less than 9.
Thus, the probability (P) is:
[tex]$ P(\text{Doubles}\ |\ \text{Sum} < 9) = \frac{\text{Number of double outcomes with sum} < 9}{\text{Total outcomes with sum} < 9} = \frac{4}{26} $[/tex]
5. Simplify and Round:
The fraction simplifies to:
[tex]$ P(\text{Doubles}\ |\ \text{Sum} < 9) = \frac{2}{13} $[/tex]
And when rounded to three decimal places, it is approximately:
[tex]$ P \approx 0.154 $[/tex]
Therefore, the probability that doubles are rolled, given that the sum on the two dice is less than 9, is [tex]\( \boxed{0.154} \)[/tex].
1. Identify Total Outcomes: When rolling two dice, each die has 6 faces. Therefore, the total number of possible outcomes when rolling two dice is [tex]\( 6 \times 6 = 36 \)[/tex].
2. Outcomes with Sum Less Than 9:
We need to determine the number of outcomes where the sum of the numbers on the two dice is less than 9. These outcomes include:
- Sum 2: (1,1)
- Sum 3: (1,2), (2,1)
- Sum 4: (1,3), (2,2), (3,1)
- Sum 5: (1,4), (2,3), (3,2), (4,1)
- Sum 6: (1,5), (2,4), (3,3), (4,2), (5,1)
- Sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
- Sum 8: (2,6), (3,5), (4,4), (5,3), (6,2)
Counting these outcomes gives us:
- 1 outcome for sum 2
- 2 outcomes for sum 3
- 3 outcomes for sum 4
- 4 outcomes for sum 5
- 5 outcomes for sum 6
- 6 outcomes for sum 7
- 5 outcomes for sum 8
Adding these up, the total number of outcomes where the sum is less than 9 is [tex]\(1 + 2 + 3 + 4 + 5 + 6 + 5 = 26\)[/tex].
3. Outcomes with Doubles and Sum Less Than 9:
Now, we need to find how many of these outcomes where the sum is less than 9 are doubles:
- For sum 4: (2,2)
- For sum 6: (3,3)
- For sum 8: (4,4)
Each of these sums has 1 double outcome:
- Total outcomes with doubles = 3
4. Calculate the Probability:
The probability that doubles are rolled, given that the sum is less than 9, is the ratio of the number of double outcomes to the total number of outcomes where the sum is less than 9.
Thus, the probability (P) is:
[tex]$ P(\text{Doubles}\ |\ \text{Sum} < 9) = \frac{\text{Number of double outcomes with sum} < 9}{\text{Total outcomes with sum} < 9} = \frac{4}{26} $[/tex]
5. Simplify and Round:
The fraction simplifies to:
[tex]$ P(\text{Doubles}\ |\ \text{Sum} < 9) = \frac{2}{13} $[/tex]
And when rounded to three decimal places, it is approximately:
[tex]$ P \approx 0.154 $[/tex]
Therefore, the probability that doubles are rolled, given that the sum on the two dice is less than 9, is [tex]\( \boxed{0.154} \)[/tex].
