To determine if events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent, we need to check if the probability of both events occurring together, [tex]\( P(A \text{ and } B) \)[/tex], equals the product of their individual probabilities, [tex]\( P(A) \)[/tex] and [tex]\( P(B) \)[/tex].
Given:
[tex]\[ P(A) = 0.60 \][/tex]
[tex]\[ P(B) = 0.30 \][/tex]
When two events are independent, the probability of both events happening together is given by:
[tex]\[ P(A \text{ and } B) = P(A) \times P(B) \][/tex]
Let's calculate that:
[tex]\[ P(A \text{ and } B) = 0.60 \times 0.30 \][/tex]
[tex]\[ P(A \text{ and } B) = 0.18 \][/tex]
Therefore, the probability of both events occurring together, assuming they are independent, is [tex]\( 0.18 \)[/tex].
Among the given options, the correct one is:
[tex]\[ D. \quad P(A \text{ and } B) = 0.18 \][/tex]
This indicates that events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent when [tex]\( P(A \text{ and } B) = 0.18 \)[/tex].
