To determine whether events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent, we need to use the definition of independent events. Two events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent if and only if
[tex]\[ P(A \cap B) = P(A) \times P(B). \][/tex]
Given:
[tex]\[ P(A) = \frac{9}{20}, \][/tex]
[tex]\[ P(B) = \frac{1}{2}, \][/tex]
[tex]\[ P(A \cap B) = \frac{9}{40}. \][/tex]
First, let's compute the product [tex]\(P(A) \times P(B)\)[/tex]:
[tex]\[ P(A) \times P(B) = \left(\frac{9}{20}\right) \times \left(\frac{1}{2}\right). \][/tex]
[tex]\[ P(A) \times P(B) = \frac{9}{20} \times \frac{1}{2} = \frac{9 \times 1}{20 \times 2} = \frac{9}{40}. \][/tex]
We see that:
[tex]\[ P(A \cap B) = \frac{9}{40} \][/tex]
and
[tex]\[ P(A) \times P(B) = \frac{9}{40}. \][/tex]
Since these two quantities are equal,
[tex]\[ P(A \cap B) = P(A) \times P(B), \][/tex]
we can conclude that events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent.
Therefore, the correct answer is:
True
