A couple has 3 children each child is equally likely to be a boy or a girl. The probability that the eldest child is a girl given that they have atleast one boy is:

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 independent events, then which of the following is/are true?

(A) $\bar{A}$ and B are independent events

(B) $P(A \cap B) = 0$

(C) $\bar{A}$ and $\bar{B}$ are independent events

(D) $P(A \cap B) = P(A) + P(B)$

Choose the correct answer from the options given below:

A random variable X has the following probability distribution:

The value of $P(X\ge 3)$ is

Three defective bulbs are mixed with $8$ good ones. If three bulbs are drawn one by one with replacement, the probabilities of getting exactly $1$ defective, more than $2$ defective, no defective and more than $1$ defective respectively are:

Let x denote the number of doublets in three throws of a pair of dice with the following probability distribution.

If value of k is equal to $\frac{m}{n} \cdot \gcd(m,n) = 1$, then $m + n$ is equal to

Let X be a discrete random variable and probability distribution X is

Then E(X) is equal to :