When two dice are thrown simultaneously,find the probability of the following:
$(1)$ The numbers are the same.
$(2)$ The difference between the numbers is $1$.

The probability of obtaining a sum of $8$ in a single throw of two dice 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 $P(A \cap \overline{B})=$

Two friends $A$ and $B$ apply for a job in the same company. The probability of $A$ getting selected is $\frac{2}{5}$ and that of $B$ is $\frac{4}{7}$. Then the probability that one of them is selected is:

From a book containing $100$ pages,one page is selected randomly. The probability that the sum of the digits of the page number of the selected page is $11$,is

An integer is chosen from the set $\{2k \mid -9 \leq k \leq 10\}$. The probability that the chosen integer is divisible by both $4$ and $6$ is