To find the probability that a randomly selected student has dark hair given that the student has brown eyes, we need to follow a series of steps. Here is a detailed, step-by-step solution:
1. Identify the relevant data:
- Number of students with brown eyes and light hair: [tex]\(9\)[/tex]
- Number of students with brown eyes and dark hair: [tex]\(11\)[/tex]
2. Calculate the total number of students with brown eyes:
[tex]\[
\text{Total students with brown eyes} = 9 \, (\text{light hair}) + 11 \, (\text{dark hair}) = 20
\][/tex]
3. Determine the number of students with dark hair among those with brown eyes:
[tex]\[
\text{Students with brown eyes and dark hair} = 11
\][/tex]
4. Calculate the probability:
- The probability that a randomly selected student has dark hair given that they have brown eyes is calculated by dividing the number of students with brown eyes and dark hair by the total number of students with brown eyes.
[tex]\[
P(\text{Dark hair} \mid \text{Brown eyes}) = \frac{\text{Students with brown eyes and dark hair}}{\text{Total students with brown eyes}} = \frac{11}{20}
\][/tex]
5. Simplify the fraction (if necessary):
- In this case, [tex]\(\frac{11}{20}\)[/tex] is already in its simplest form and cannot be simplified further.
6. Express the result as a decimal (if needed):
[tex]\[
\frac{11}{20} = 0.55
\][/tex]
So, the probability that a randomly selected student has dark hair given that the student has brown eyes is [tex]\(\boxed{0.55}\)[/tex].
