$A$ black and a red dice are rolled. Find the conditional probability of obtaining a sum greater than $9,$ given that the black die resulted in a $5$.

Let $A$ and $B$ be two non-null events such that $A \subset B$. Then,which of the following statements is always correct?

Let $A$ and $B$ be two independent events such that $P(A) + P(B) = \frac{3}{4}$ and $P(\overline{A} | B) = \frac{2}{5}$. Then,$P(A \cap B)$ is -

$A$ box contains $3$ white and $2$ red balls. One ball is drawn at random,and then a second ball is drawn without replacement. What is the probability that the second ball is red?

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 $3$'.

$A$ bag contains $4$ Red and $6$ Black balls. $A$ ball is drawn at random from the bag,its colour is observed and this ball along with $3$ additional balls of the same colour are returned to the bag. If now a ball is drawn at random from the bag,then the probability that this drawn ball is red is