Answered

If [tex]$P(A)=\frac{9}{20}$[/tex], [tex]$P(B)=\frac{1}{2}$[/tex], and [tex]$P(A \text{ and } B)=\frac{9}{40}$[/tex], then [tex]$A$[/tex] and [tex]$B$[/tex] are independent events.

True
False



Answer :

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