Answer :
Let's go through the problem step-by-step to determine the correct answer.
1. Total number of pencils in the pencil case:
Bonnie has [tex]\(4\)[/tex] sharpened and [tex]\(8\)[/tex] unsharpened pencils in her pencil case.
Therefore, the total number of pencils is:
[tex]\[ 4 + 8 = 12 \][/tex]
2. Probability of first pencil being sharpened:
The probability that the first pencil selected is sharpened is the ratio of the number of sharpened pencils to the total number of pencils:
[tex]\[ \frac{4}{12} = \frac{1}{3} \][/tex]
3. Probability of second pencil being sharpened given that the first one was sharpened:
If the first pencil selected is sharpened, then there are now [tex]\(3\)[/tex] sharpened pencils left out of the remaining [tex]\(11\)[/tex] pencils.
Therefore, the probability that the second pencil is also sharpened is:
[tex]\[ \frac{3}{11} \][/tex]
4. Overall probability that both pencils are sharpened:
The overall probability that both pencils selected are sharpened is the product of the two individual probabilities calculated above:
[tex]\[ \left(\frac{1}{3}\right) \times \left(\frac{3}{11}\right) = \frac{1 \times 3}{3 \times 11} = \frac{3}{33} = \frac{1}{11} \][/tex]
Therefore, the probability that both pencils will be sharpened is [tex]\(\frac{1}{11}\)[/tex].
So, the correct answer is:
[tex]\[ \boxed{\frac{1}{11}} \][/tex]
1. Total number of pencils in the pencil case:
Bonnie has [tex]\(4\)[/tex] sharpened and [tex]\(8\)[/tex] unsharpened pencils in her pencil case.
Therefore, the total number of pencils is:
[tex]\[ 4 + 8 = 12 \][/tex]
2. Probability of first pencil being sharpened:
The probability that the first pencil selected is sharpened is the ratio of the number of sharpened pencils to the total number of pencils:
[tex]\[ \frac{4}{12} = \frac{1}{3} \][/tex]
3. Probability of second pencil being sharpened given that the first one was sharpened:
If the first pencil selected is sharpened, then there are now [tex]\(3\)[/tex] sharpened pencils left out of the remaining [tex]\(11\)[/tex] pencils.
Therefore, the probability that the second pencil is also sharpened is:
[tex]\[ \frac{3}{11} \][/tex]
4. Overall probability that both pencils are sharpened:
The overall probability that both pencils selected are sharpened is the product of the two individual probabilities calculated above:
[tex]\[ \left(\frac{1}{3}\right) \times \left(\frac{3}{11}\right) = \frac{1 \times 3}{3 \times 11} = \frac{3}{33} = \frac{1}{11} \][/tex]
Therefore, the probability that both pencils will be sharpened is [tex]\(\frac{1}{11}\)[/tex].
So, the correct answer is:
[tex]\[ \boxed{\frac{1}{11}} \][/tex]
