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[tex]$P(A)=\frac{3}{5}, P(B)=\frac{1}{4}$[/tex], and [tex]$P(A \cap B)=\frac{3}{10}$[/tex]. Are [tex]$A$[/tex] and [tex]$B$[/tex] independent?

Select the option that provides both the correct answer and the correct reason.

A. Yes, because [tex]$\frac{3}{5}\left(\frac{1}{4}\right)=\frac{3}{10}$[/tex].
B. No, because [tex]$\frac{3}{5}\left(\frac{1}{4}\right) \neq \frac{3}{10}$[/tex].
C. Yes, because [tex]$\frac{3}{5}+\frac{1}{4}=\frac{3}{10}$[/tex].
D. No, because [tex]$\frac{3}{5} \div \frac{1}{4} \neq \frac{3}{10}$[/tex].
E. No, because [tex]$\frac{3}{5}+\frac{1}{4} \neq \frac{3}{10}$[/tex].

Sagot :

GinnyAnsweridn
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].
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