If two subsets $A$ and $B$ are selected at random from a set $S$ containing $n$ elements,then the probability that $A \cap B = \phi$ and $A \cup B = S$ is

Two dice are thrown and the sum of the numbers appeared on the dice is noted. If $A$ is the event of getting a prime number as their sum and $B$ is the event of getting a number greater than $8$ as their sum,then $P(A \cap \overline{B})=$

Let $A$ and $B$ be events in a sample space $S$ such that $P(A)=0.5, P(B)=0.4$ and $P(A \cup B)=0.6$. Observe the following lists. The correct match of List $I$ from List $II$ is:

List $I$	List $II$
$(i) \ P(A \cap B)$	$(1) \ 0.4$
$(ii) \ P(A \cap \bar{B})$	$(2) \ 0.2$
$(iii) \ P(\bar{A} \cap B)$	$(3) \ 0.3$
$(iv) \ P(\bar{A} \cap \bar{B})$	$(4) \ 0.1$

$A$ problem in statistics is given to three students $A, B$ and $C$. Their probabilities of solving the problem are $\frac{1}{2}, \frac{1}{3}$ and $\frac{1}{4}$ respectively. If all of them try independently,then the probability that the problem is solved is

One card is drawn from a well-shuffled deck of $52$ cards. If each outcome is equally likely,calculate the probability that the card will be a diamond but not an ace.

For two events $A$ and $B$,if $P(A \cup B) = \frac{3}{4}$,$P(A \cap B) = \frac{1}{4}$,and $P(A') = \frac{2}{3}$,then find $P(A' \cap B)$.