Suppose that A and B are independent events with $P(A)=\dfrac{1}{7}$ and $P(B)=\dfrac{3}{7}$, then $P(A\cap B)$ is

The probability distribution table of a random variable X is given below.

If $E(X) = \dfrac{3}{2}$, then the values of a and b are:

It is given that student marked the answer correctly, the probability that he guesses is

The probability distribution of a random variable x is given below.

Then the value of k is

An alphabet has been selected from English alphabets. What is the probability that it is a vowel given that it precedes 'M' ?

Let X denote the number of hours a student studies on a selected day. The probability distribution of X is given by (where k is some unknown constant)

$P(X = x_i) = \begin{cases} 0.5, & \text{if } x_i = 0, \\ kx_i, & \text{if } x_i = 1, \\ k(4 - x_i), & \text{if } x_i = 2 \text{ or } 3, \\ 0, & \text{otherwise}. \end{cases}$

Then the value of k is

If A and B are two events such that $P(A) = \frac{1}{2}$, $P(B) = \frac{1}{3}$ and $P(A \cap B) = \frac{1}{4}$, then which of the following statements are true?

(A) A and B are independent events

(B) $P(A | B) = \frac{3}{4}$

(C) $P(A' | B') = \frac{5}{8}$

(D) $P(A' | B) = \frac{1}{4}$

Choose the correct answer from the options given below: