If two smallest squares are chosen at random on a chessboard,then the probability of choosing these squares such that they do not have a side in common is:

Two dice are thrown and two coins are tossed simultaneously. The probability of getting prime numbers on both the dice along with a head and a tail on the two coins is

If $A$ and $B$ are two independent events such that $P(B)=\frac{2}{7}$ and $P\left(A \cup B^c\right)=0.8$,then $P(A \cup B)$ $=$

Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are the same. If the probability of a random toss resulting in a head is $\frac{1}{3}$,then the probability that the experiment stops with heads is:

There are four machines and it is known that exactly two of them are faulty. They are tested one by one,in a random order until both the faulty machines are identified. The probability that only two tests are needed is:

Let $X$ and $Y$ be two events such that $P(X \mid Y)=\frac{1}{2}$,$P(Y \mid X)=\frac{1}{3}$,and $P(X \cap Y)=\frac{1}{6}$. Which of the following is (are) correct?
$(A)$ $P(X \cup Y)=\frac{2}{3}$
$(B)$ $X$ and $Y$ are independent
$(C)$ $X$ and $Y$ are not independent
$(D)$ $P(X^C \cap Y)=\frac{1}{3}$