If $A$ and $B$ are two events such that $P ( A \cap B ) = 0.1$, and $P ( A \mid B )$ and $P ( B \mid A )$ are the roots of the equation $12 x ^ { 2 } - 7 x + 1 = 0$, then the value of $\frac { \mathrm { P } ( \overline { \mathrm { A } } \cup \overline { \mathrm { B } } ) } { \mathrm { P } ( \overline { \mathrm { A } } \cap \overline { \mathrm { B } } ) }$ is :
(1) $\frac { 4 } { 3 }$
(2) $\frac { 7 } { 4 }$
(3) $\frac { 5 } { 3 }$
(4) $\frac { 9 } { 4 }$

On a table, there are three marbles in total: one red, one blue, and one yellow. These marbles are placed in bags A, B, and C with one marble in each bag, and p: ``There is no red marble in bag A.'' q: ``There is a blue marble in bag B.'' r: ``There is no yellow marble in bag C.'' propositions are given.
$$p \wedge ( q \vee r ) ^ { \prime \prime }$$
Given that the proposition is true; what are the colors of the marbles in bags A, B and C respectively?
A) Red - Blue - Yellow
B) Blue - Red - Yellow
C) Blue - Yellow - Red
D) Yellow - Red - Blue
E) Yellow - Blue - Red

Regarding the subsets $A, B$ and $C$ of the set of natural numbers, the propositions
$$\begin{aligned} & p : 9 \in A \cup B \\ & q : 9 \in A \cap C \\ & r : 9 \notin C \end{aligned}$$
are given. Given that the proposition $(p \Rightarrow q)' \wedge r'$ is true, which of the following statements are true?
I. $9 \in A$ II. $9 \in B$ III. $9 \in C$
A) Only I B) Only III C) I and II D) II and III E) I, II and III