To determine if events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent, we need to check if [tex]\(P(A \cap B) = P(A) \cdot P(B)\)[/tex].
Given:
- [tex]\(P(A) = \frac{3}{5}\)[/tex]
- [tex]\(P(B) = \frac{1}{4}\)[/tex]
- [tex]\(P(A \cap B) = \frac{3}{10}\)[/tex]
Let's calculate [tex]\(P(A) \cdot P(B)\)[/tex]:
[tex]\[ P(A) \cdot P(B) = \left(\frac{3}{5}\right) \cdot \left(\frac{1}{4}\right) \][/tex]
Multiply the fractions by multiplying the numerators and the denominators:
[tex]\[ P(A) \cdot P(B) = \frac{3 \times 1}{5 \times 4} = \frac{3}{20} \][/tex]
Next, compare [tex]\(P(A \cap B)\)[/tex] with [tex]\(P(A) \cdot P(B)\)[/tex]:
[tex]\[ \frac{3}{10} \neq \frac{3}{20} \][/tex]
Since [tex]\(P(A) \cdot P(B) \neq P(A \cap B)\)[/tex], events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are not independent.
Therefore, the correct option is:
No, because [tex]\(\frac{3}{5}\left(\frac{1}{4}\right) \neq \frac{3}{10}\)[/tex].
