Statistics and Probability

Two fair dice are thrown simultaneously. The probability that the sum of the numbers shown is $7$ is

Given three identical bags each containing 10 balls, whose colours are as follows: \\ \textbf{Bag I:} Red = 3, Blue = 2, Green = 5 \\ \textbf{Bag II:} Red = 4, Blue = 3, Green = 3 \\ \textbf{Bag III:} Red = 5, Blue = 1, Green = 4 \\ A person chooses a bag at random and takes out a ball. If the ball is Red, the probability that it is from bag I is $p$ and if the ball is Green, the probability that it is from bag III is $q$, then the value of $\left( \frac{1}{p} + \frac{1}{q} \right)$ is:

If the mean and the variance of $6,\ 4,\ a,\ 8,\ b,\ 12,\ 10,\ 13$ are $9$ and $9.25$ respectively, then $a + b + ab$ is equal to:

Let the mean and variance of five observations $x_1=1$, $x_2=3$, $x_3=a$, $x_4=7$ and $x_5=b$, with $a>b$, be $5$ and $10$ respectively. Then the variance of the observations $n + x_n$ for $n=1,2,\dots,5$ is:

If the probability that the random variable $X$ takes the value $x$ is given by $P(X=x)=k\,(x+1)3^{-x},\quad x=0,1,2,\dots,$ where $k$ is a constant, then $P(X\ge3)$ is equal to:

Let the mean and the standard deviation of the observations 2, 3, 3, 4, 5, 7, a, b be 4 and √2 respectively. Then the mean deviation about the mode of these observations is:

The probability of forming a 12-person committee from 4 engineers, 2 doctors and 10 professors containing at least 3 engineers and at least 1 doctor is:

A box contains 10 pens of which 3 are defective. A sample of 2 pens is drawn at random and let X denote the number of defective pens. Then the variance of X is:

A bag contains 19 unbiased coins and one coin with heads on both sides. One coin drawn at random is tossed and a head turns up. If the probability that the drawn coin was unbiased is $\frac{m}{n}$, with $\gcd(m,n)=1$, then $n^2 - m^2$ is equal to:

Let a random variable X take values 0, 1, 2 and 3 with P(X = 0) = P(X = 1) = p and P(X = 2) = P(X = 3) = $\tfrac{1 - 2p}{2}$. If $E(X^2) = 2E(X)$, then the value of $8p - 1$ is: