Consider the experiment of throwing a die. If a multiple of $3$ comes up,throw the die again,and if any other number comes,toss a coin. Find the conditional probability of the event 'the coin shows a tail',given that 'at least one die shows a $3$'.

$A$ coin is tossed three times. If event $E$ represents getting at least two heads and event $F$ represents getting a head on the first toss,find $P(E|F)$.

$A$ and $B$ are two groups of books. Group $A$ consists of $8$ science and $5$ engineering books, and group $B$ consists of $6$ science and $7$ engineering books. When an unbiased die is rolled, if $2$ or $5$ turns up, a book is selected at random from group $A$. Otherwise, a book is selected at random from group $B$. The probability of selecting a science book is

Let $A$ and $B$ be two independent events such that $P(A)=\frac{1}{3}$ and $P(B)=\frac{1}{6}$. Then,which of the following is $TRUE$?

Let $E^c$ denote the complement of an event $E$. Let $E, F, G$ be pairwise independent events with $P(G)>0$ and $P(E \cap F \cap G) = 0$. Then $P(E^c \cap F^c \mid G)$ equals

An experiment has $10$ equally likely outcomes. Let $A$ and $B$ be two non-empty events of the experiment. If $A$ consists of $4$ outcomes,the number of outcomes that $B$ must have so that $A$ and $B$ are independent,is