To determine the conditional probability [tex]\( P(B \mid A) \)[/tex], we use the following formula:
[tex]\[
P(B \mid A) = \frac{P(A \cap B)}{P(A)}
\][/tex]
Given the probabilities:
[tex]\[
P(A) = \frac{13}{20}
\][/tex]
[tex]\[
P(A \cap B) = \frac{13}{25}
\][/tex]
We substitute these values into the formula:
[tex]\[
P(B \mid A) = \frac{P(A \cap B)}{P(A)} = \frac{\frac{13}{25}}{\frac{13}{20}}
\][/tex]
To simplify this division of fractions, we multiply the numerator by the reciprocal of the denominator:
[tex]\[
P(B \mid A) = \frac{13}{25} \times \frac{20}{13}
\][/tex]
The [tex]\(13\)[/tex] in the numerator and the [tex]\(13\)[/tex] in the denominator cancel each other out:
[tex]\[
P(B \mid A) = \frac{20}{25}
\][/tex]
Then we simplify [tex]\(\frac{20}{25}\)[/tex]:
[tex]\[
P(B \mid A) = \frac{4}{5}
\][/tex]
Therefore, the conditional probability [tex]\( P(B \mid A) \)[/tex] is:
[tex]\[
\boxed{\frac{4}{5}}
\][/tex]
Thus, the correct answer is:
A. [tex]\(\frac{4}{5}\)[/tex]
