Find the missing probability.

[tex]\[ P(A)=\frac{13}{20}, \quad P(A \cap B)=\frac{13}{25}, \quad P(B \mid A)=? \][/tex]

A. [tex]\(\frac{21}{100}\)[/tex]
B. [tex]\(\frac{1}{5}\)[/tex]
C. [tex]\(\frac{1}{4}\)[/tex]
D. [tex]\(\frac{4}{5}\)[/tex]



Answer :

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]