To find the missing probability [tex]\( P(B \mid A) \)[/tex], we use the concept of conditional probability. Conditional probability is the probability of an event occurring given that another event has already occurred. The formula for conditional probability [tex]\( P(B \mid A) \)[/tex] is given by:
[tex]\[
P(B \mid A) = \frac{P(A \cap B)}{P(A)}
\][/tex]
Here, we are given:
- [tex]\( P(A) = \frac{13}{20} \)[/tex]
- [tex]\( P(A \cap B) = \frac{13}{25} \)[/tex]
Now, we substitute these values into the conditional probability formula:
[tex]\[
P(B \mid A) = \frac{P(A \cap B)}{P(A)} = \frac{\frac{13}{25}}{\frac{13}{20}}
\][/tex]
To simplify this expression, we can multiply by the reciprocal of the denominator:
[tex]\[
P(B \mid A) = \frac{13}{25} \times \frac{20}{13}
\][/tex]
When we multiply the fractions, the 13s cancel out:
[tex]\[
P(B \mid A) = \frac{20}{25}
\][/tex]
Simplifying the fraction:
[tex]\[
P(B \mid A) = \frac{4}{5}
\][/tex]
Thus, the conditional probability [tex]\( P(B \mid A) \)[/tex] is [tex]\( \frac{4}{5} \)[/tex].
Therefore, the correct answer is:
D. [tex]\( \frac{4}{5} \)[/tex]
