To solve the problem, we need to calculate the probability that a randomly selected student from the class plays volleyball given that they play basketball. This specific probability is known as a conditional probability and can be found using the formula for conditional probability.
We are given:
- The total number of students in the class: [tex]\(30\)[/tex]
- The number of students who play basketball: [tex]\(18\)[/tex]
- The number of students who play volleyball: [tex]\(12\)[/tex]
- The number of students who play both basketball and volleyball: [tex]\(9\)[/tex]
Let us denote the following events:
- [tex]\(B\)[/tex]: The event that a student plays basketball.
- [tex]\(V\)[/tex]: The event that a student plays volleyball.
We need to find the conditional probability [tex]\(P(V|B)\)[/tex], which is the probability that a student plays volleyball given that they play basketball. The formula for this conditional probability is:
[tex]\[
P(V|B) = \frac{P(V \cap B)}{P(B)}
\][/tex]
Here:
- [tex]\(P(V \cap B)\)[/tex] is the probability that a student plays both volleyball and basketball.
- [tex]\(P(B)\)[/tex] is the probability that a student plays basketball.
First, we calculate [tex]\(P(B)\)[/tex]:
[tex]\[
P(B) = \frac{\text{Number of students who play basketball}}{\text{Total number of students}} = \frac{18}{30} = \frac{3}{5}
\][/tex]
Next, we calculate [tex]\(P(V \cap B)\)[/tex]:
[tex]\[
P(V \cap B) = \frac{\text{Number of students who play both basketball and volleyball}}{\text{Total number of students}} = \frac{9}{30} = \frac{3}{10}
\][/tex]
Now, we substitute these values into the conditional probability formula:
[tex]\[
P(V|B) = \frac{P(V \cap B)}{P(B)} = \frac{\frac{3}{10}}{\frac{3}{5}} = \frac{3}{10} \times \frac{5}{3} = \frac{1}{2}
\][/tex]
Therefore, the probability that a randomly selected student from the class plays volleyball given that they play basketball is:
[tex]\[
\boxed{\frac{1}{2}}
\][/tex]
So, there is a [tex]\(50\%\)[/tex] chance that a student who plays basketball also plays volleyball.
