If $A$ and $B$ are two events such that $P(A \cup B) = P(A \cap B)$,then the true relation is

$A$ die is thrown three times. Events $A$ and $B$ are defined as below:
$A$: $4$ on the third throw
$B$: $6$ on the first and $5$ on the second throw
Find the probability of $A$ given that $B$ has already occurred.

Consider the following statements.
Statement $(I)$: If $E$ and $F$ are two independent events,then $E^{\prime}$ and $F^{\prime}$ are also independent.
Statement $(II)$: Two mutually exclusive events with non-zero probabilities of occurrence cannot be independent.
Which of the following is correct?

Find $P(E | F)$ for a coin tossed three times,where $E$ is the event 'at least two heads' and $F$ is the event 'at most two heads'.

$A$ coin is tossed until a head appears or it has been tossed thrice. Given that a head does not appear on the first toss,what is the probability that the coin is tossed thrice?

In a city,$40\%$ of the people have brown hair,$25\%$ have brown eyes,and $15\%$ have both brown hair and brown eyes. If a person is selected at random from those having brown hair,what is the probability that they also have brown eyes?