If any four numbers are selected,what is the probability that the last digit of their product will be $1, 3, 5$ or $7$?

The probability that Ajay will not appear in the $JEE$ exam is $p = \frac{2}{7}$,while the probability that both Ajay and Vijay will appear in the exam is $q = \frac{1}{5}$. Then the probability that Ajay will appear in the exam and Vijay will not appear is:

Two dice are thrown together. The probability that the sum of the two numbers will be a multiple of $4$ is

Let $A$ and $B$ be events with $P(A)=\frac{1}{3}, P(B)=\frac{1}{4}$ and $P(A \cup B)=\frac{1}{2}$. Then which of the following statements is incorrect?

Let $M$ be the maximum value of the product of two positive integers when their sum is $66$. Let the sample space $S = \{x \in \mathbb{Z} : x(66 - x) \geq \frac{5}{9} M\}$ and the event $A = \{x \in S : x \text{ is a multiple of } 3\}$. Then $P(A)$ is equal to

$A$ die is rolled. Let $E$ be the event "die shows $4$" and $F$ be the event "die shows an even number". Are $E$ and $F$ mutually exclusive?