If $A$ and $B$ are events,such that $P(A) = \frac{1}{4}$,$P(A|B) = \frac{1}{2}$,and $P(B|A) = \frac{2}{3}$,then $P(B)$ is

If $A$ and $B$ are two events such that $P(A) = \frac{2}{3}$, $P(B) = \frac{1}{2}$ and $P(A | B) = \frac{2}{3}$, then $P(A' \cup B) + P(A \cup B') = $

Prove that if $E$ and $F$ are independent events,then so are the events $E$ and $F^{\prime}$.

If $A$ and $B$ are any two events such that $P(A) = \frac{2}{5}$ and $P(A \cap B) = \frac{3}{20}$,then the conditional probability $P(A | A' \cup B')$,where $A'$ denotes the complement of $A$,is equal to:

An unbiased die is thrown twice. Let the event $A$ be 'odd number on the first throw' and $B$ the event 'odd number on the second throw'. Check the independence of the events $A$ and $B$.

$A$ coin is tossed three times in succession. If $E$ is the event that there are at least two heads and $F$ is the event in which the first throw is a head,then $P(E|F) = $