If $P\,(A) = 0.4,\,\,P\,(B) = x,\,\,P\,(A \cup B) = 0.7$ and the events $A$ and $B$ are independent, then $x =$

Let $A$ and $B $ be two events such that  $P\left( {\overline {A \cup B} } \right) = \frac{1}{6}\;,P\left( {A \cap B} \right) = \frac{1}{4}$ and $P\left( {\bar A} \right) = \frac{1}{4}$ where $\bar A$ stands for the complement of the event $A$. Then the events $A$ and$B$ are

Let $X$ and $Y$ are two events such that $P(X \cup Y=P)\,(X \cap Y).$

Statement $1:$ $P(X \cap Y' = P)\,(X' \cap Y = 0).$

Statement $2:$ $P(X) + P(Y = 2)\,P\,(X \cap Y)$

If $A$ and $B$ are two events, then the probability of the event that at most one of $A, B$ occurs, is

An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the numbers obtained on the is noted. If the toss of the coin results in tail then a card from a well-shuffled pack of nine cards numbered $1, 2, 3,….., 9$ is randomly picked and the number on the card is noted. The probability that the noted number is either $7$ or $8$ is