If events $A$ and $B$ are such that $P(A) = \frac{1}{4}$, $P(A \cap B) = \frac{1}{10}$ and $P(A|B) = 2P(B|A)$, then $P(B) = ..........$

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

Let $A, B$ and $C$ be three events,which are pairwise independent and $\bar{E}$ denotes the complement of an event $E$. If $P(A \cap B \cap C) = 0$ and $P(C) > 0$,then $P[(\bar{A} \cap \bar{B})|C]$ is equal to

Four fair dice $D_1, D_2, D_3$ and $D_4$,each having six faces numbered $1, 2, 3, 4, 5$ and $6$,are rolled simultaneously. The probability that $D_4$ shows a number appearing on at least one of $D_1, D_2$ and $D_3$ is

If $\overline{E}$ and $\overline{F}$ are the complementary events of events $E$ and $F$ respectively and if $0 < P(F) < 1$,then

$A$ 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.