To solve for [tex]\( P(A \cap B) \)[/tex], we can use the definition of conditional probability and the multiplication rule for dependent events.
We are given:
- [tex]\( P(A) = 0.5 \)[/tex]
- [tex]\( P(B \mid A) = 0.6 \)[/tex]
The formula that relates these probabilities is:
[tex]\[ P(A \cap B) = P(B \mid A) \times P(A) \][/tex]
Plugging in the values provided:
[tex]\[ P(A \cap B) = 0.6 \times 0.5 \][/tex]
Now, multiply the two probabilities:
[tex]\[ P(A \cap B) = 0.3 \][/tex]
Thus, the value of [tex]\( P(A \cap B) \)[/tex] is [tex]\( 0.3 \)[/tex].
So the correct answer is:
[tex]\[ \boxed{0.3} \][/tex]
