To determine the probability of both events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] occurring together, [tex]\( P(A \cap B) \)[/tex], for dependent events, we use the definition of conditional probability.
The formula for the probability of the intersection of two dependent events is given by:
[tex]\[ P(A \cap B) = P(A) \times P(B \mid A) \][/tex]
Here, we are given:
- [tex]\( P(A) = 0.5 \)[/tex]
- [tex]\( P(B \mid A) = 0.6 \)[/tex]
Substituting these values into the formula, we get:
[tex]\[ P(A \cap B) = 0.5 \times 0.6 \][/tex]
Now, multiplying these together:
[tex]\[ P(A \cap B) = 0.3 \][/tex]
Therefore, the probability that both events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] occur is [tex]\( 0.3 \)[/tex].
So the correct answer is
[tex]\[
\boxed{0.3}
\][/tex]
which corresponds to option [tex]\( D \)[/tex].
