To solve the problem of finding [tex]\( P(A \mid B) \)[/tex] given that [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent events, we follow these steps:
1. Understanding Independence:
When two events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent, the occurrence of one event does not affect the probability of the other. Mathematically, independence between events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is expressed as:
[tex]\[
P(A \cap B) = P(A) \cdot P(B)
\][/tex]
2. Conditional Probability:
The conditional probability [tex]\( P(A \mid B) \)[/tex] represents the probability of event [tex]\( A \)[/tex] occurring given that event [tex]\( B \)[/tex] has occurred. The formula for conditional probability is:
[tex]\[
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
\][/tex]
3. Applying Independence:
Since [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent, we know that:
[tex]\[
P(A \cap B) = P(A) \cdot P(B)
\][/tex]
Using this in the conditional probability formula, we get:
[tex]\[
P(A \mid B) = \frac{P(A) \cdot P(B)}{P(B)}
\][/tex]
4. Simplification:
In the numerator and the denominator, [tex]\( P(B) \)[/tex] cancels out:
[tex]\[
P(A \mid B) = \frac{P(A) \cdot P(B)}{P(B)} = P(A)
\][/tex]
Therefore, for independent events, the conditional probability [tex]\( P(A \mid B) \)[/tex] is just [tex]\( P(A) \)[/tex].
5. Given Probabilities:
From the problem, we know that:
[tex]\[
P(A) = 0.60
\][/tex]
So, the conditional probability [tex]\( P(A \mid B) \)[/tex] is:
[tex]\[
P(A \mid B) = P(A) = 0.60
\][/tex]
Thus, the answer is:
[tex]\[
\boxed{0.60}
\][/tex]
Therefore, the correct option is:
C. 0.60
