Two dice are thrown together. If the numbers appearing on the two dice are different,then what is the probability that the sum is $6$?

Two dice are thrown. The events $A, B$ and $C$ are as follows:
$A:$ getting an even number on the first die.
$B:$ getting an odd number on the first die.
$C:$ getting the sum of the numbers on the dice $\leq 5$.
State true or false: (give reason for your answer)
Statement: $A', B', C$ are mutually exclusive and exhaustive.

Let $S$ be the sample space of all five-digit numbers. If $p$ is the probability that a randomly selected number from $S$ is a multiple of $7$ but not divisible by $5$,then $9p$ is equal to.

The probability of getting head and tail alternately in three throws of a coin (or a throw of three coins) is:

$A$ problem in Algebra is given to two students $A$ and $B$ whose chances of solving it are $\frac{2}{5}$ and $\frac{3}{4}$ respectively. The probability that the problem is solved, if both of them try independently, is

Let $A, B$ and $C$ be three events associated with sample space $S$. $A, B$ and $C$ are pairwise independent and $P(A)=P(B)=P(C)=P$. If all of them cannot occur simultaneously,then $P(A \cup B \cup C)$ is equal to