Three bags contain a number of red and white balls as follows:

Bag I: 3 red balls

Bag II: 2 red balls and 1 white ball

Bag III: 3 white balls

The probability that bag $i$ will be chosen, and a ball is selected from it is $\frac{i}{6}, i=1,2,3$. The probability that a red ball is selected is equal to:

If a random variable y follows Poisson's distribution such that $P(X = 2) = 9 P(X = 4) + 90 P(X = 6)$, then sum of the mean and variance of X is

Suppose X has Poisson distribution such that $3 P(X=1) = 2 P(X=2)$ then $P(X>0)$ is:

If the binomial distribution $X\sim B(n, p)$ of mean 3 and variance $\frac{3}{2}$, $(p + q) = 1$, then which of the following is/are TRUE?

(A) $q = \frac{1}{2}$, $n = 6$

(B) $P(X \leq 5) = \frac{63}{64}$, $p = \frac{1}{2}$

(C) $q = \frac{1}{3}$, $p = \frac{2}{3}$

(D) $P(X = 4) = \frac{15}{64}$, $n = 6$

Choose the correct answer from the options given below:

The random variable $x$ has a probability distribution $p(x)$ of the form $P(x=r)=\begin{cases} rk, & \text{if } r \le 2, \\ (r-1)k, & \text{if } 2<r\le 4, \\ 0, & \text{otherwise,} \end{cases}$ where $r \in \mathbb{N}\cup\{0\}$ and $k\in\mathbb{R}$, where $\mathbb{N}$ is the set of natural numbers. Then:

(A) $k=\frac{1}{9}$

(B) $P(2\le x\le 3)=\frac{1}{2}$

(C) $P(x=4)=\frac{1}{3}$

(D) $P(x>1)=\frac{7}{8}$

Choose the correct answer from the options given below.

The random variable X has the following probability distribution:

The correct value of k is:

If E and F are two events such that $P(E \cup F)=P(E)$, then