For two events $A$ and $B$,if $P(A) = P(A|B) = \frac{1}{4}$ and $P(B|A) = \frac{1}{2}$,then:

In a school,there are $1000$ students,out of which $430$ are girls. It is known that out of $430$,$10\%$ of the girls study in class $XII$. What is the probability that a student chosen randomly studies in class $XII$ given that the chosen student is a girl?

Let $A$ and $B$ be independent events with $P(B) = \frac{2}{5}$ and $P(A \cup B) = \frac{11}{20}$. Then $P(A' \mid B)$ is a root of which equation?

Find $P(E | F)$ for a coin tossed three times,where $E: \text{head on third toss}$,$F: \text{heads on first two tosses}$.

Two dice are thrown simultaneously. If $A$ is the event of getting the sum of the numbers on two dice as greater than or equal to $8$ and $B$ is the event of getting a number less than or equal to $3$ on at least one of the dice. Then,$P(B / A) = $

If $A$ and $B$ are two events such that $P(A | B) = 0.6$, $P(B | A) = 0.3$, and $P(A) = 0.1$, then $P(\bar{A} \cap \bar{B})$ equals: