The probability of solving a problem by three persons $A, B$ and $C$ independently is $\frac{1}{2}, \frac{1}{4}$ and $\frac{1}{3}$ respectively. Then the probability that the problem is solved by exactly two of them is

Three coins are tossed. Describe two events which are mutually exclusive but not exhaustive.

For independent events $A_1, A_2, \dots, A_n$,$P(A_i) = \frac{1}{i + 1}$ for $i = 1, 2, \dots, n$. Then the probability that none of the events will occur is:

$A$ die has two faces each with number $1$,three faces each with number $2$,and one face with number $3$. If the die is rolled once,determine $P(\text{not } 3)$.

Let $S$ be a set of $5$ elements and $P(S)$ denote the power set of $S$. Let $E$ be an event of choosing an ordered pair $(A, B)$ from the set $P(S) \times P(S)$ such that $A \cap B = \varnothing$. If the probability of the event $E$ is $\frac{3^p}{2^q}$, where $p, q \in N$, then $p+q$ is equal to

From the word $POSSESSIVE$,a letter is chosen at random. The probability of it being $S$ is