IDNLearn.com provides a platform for sharing and gaining valuable knowledge. Our experts are available to provide accurate, comprehensive answers to help you make informed decisions about any topic or issue you encounter.
Sagot :
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].
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].
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. IDNLearn.com has the solutions to your questions. Thanks for stopping by, and come back for more insightful information.