Let $A$ and $E$ be any two events with positive probabilities Statement I: $P ( E / A ) \geq P ( A / E ) P ( E )$. Statement II: $P ( A / E ) \geq P ( A \cap E )$.
(1) Both the statements are false
(2) Both the statements are true
(3) Statement-I is false, Statement-II is true
(4) Statement - I is true, Statement - II is false

Let $A$ and $B$ be two non-null events such that $A \subset B$. Then, which of the following statements is always correct?
(1) $P(A \mid B) \geq P(A)$
(2) $P(A \mid B) = P(B) - P(A)$
(3) $P(A \mid B) \leq P(A)$
(4) $P(A \mid B) = 1$

Let $A$ and $B$ be two events such that $P ( B \mid A ) = \frac { 2 } { 5 } , P ( A \mid B ) = \frac { 1 } { 7 }$ and $P ( A \cap B ) = \frac { 1 } { 9 }$. Consider $( S1 )\; P \left( A ^ { \prime } \cup B \right) = \frac { 5 } { 6 }$, $( S2 )\; P \left( A ^ { \prime } \cap B ^ { \prime } \right) = \frac { 1 } { 18 }$. Then
(1) Both $(S1)$ and $(S2)$ are true
(2) Both $(S1)$ and $(S2)$ are false
(3) Only $(S1)$ is true
(4) Only $(S2)$ is true

Let Ajay will not appear in JEE exam with probability $p = \frac{2}{7}$, while both Ajay and Vijay will appear in the exam with probability $q = \frac{1}{5}$. Then the probability, that Ajay will appear in the exam and Vijay will not appear is:
(1) $\frac{9}{35}$
(2) $\frac{18}{35}$
(3) $\frac{24}{35}$
(4) $\frac{3}{35}$

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) $\frac { 9 } { 4 }$

Knowing that $P ( A ) = 0.5 , P ( A / B ) = 0.625$ and $P ( A \cup B ) = 0.65$, find:\ a) ( 1.5 points) $P ( B )$ and $P ( A \cap B )$.\ b) (1 point) $P ( A / A \cup B )$ and $P ( A \cap B / A \cup B )$

9. Consider the statement about Fred: (}) Every day next week, Fred will do at least one maths problem. If statement (*) is not true, which of the following is certainly true?
A Every day next week, Fred will do more than one maths problem.
B Some day next week, Fred will do more than one maths problem.
C On no day next week will Fred do more than one maths problem.
D Every day next week, Fred will do no maths problems.
E Some day next week, Fred will do no maths problems. F On no day next week will Fred do no maths problems.

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