Probability & Statistics PYQ — Page 10

Two dices are rolled. If both dices have six faces numbered $1,2,3,5,7$ and $11,$ then the probability that the sum of the numbers on the top faces is less than or equal to $8$ is:

Let $A$ be a set of all $4$ -digit natural numbers whose exactly one digit is $7$. Then the probability that a randomly chosen element of $A$ leaves remainder $2$ when divided by $5$ is:

Consider three observations $a,b$ and $c$ such that $b=a+c$. If the standard deviation of $a+2,c+2$ is $d$, then which of the following is true?

In a group of $400$ people, $160$ are smokers and non-vegetarian; $100$ are smokers and vegetarian and the remaining $140$ are non-smokers and vegetarian. Their chances of getting a particular chest disorder are $35%,20%$ and $10%$ respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is :

Consider a set of $3n$ numbers having variance $4.$ In this set, the mean of first $2n$ numbers is $6$ and the mean of the remaining $n$ numbers is $3.$ A new set is constructed by adding $1$ into each of the first $2n$ numbers, and subtracting $1$ from each of the remaining $n$ numbers. If the variance of the new set is $k,$ then $9k$ is equal to ______.

A student appeared in an examination consisting of $8$ true-false type questions. The student guesses the answers with equal probability. The smallest value of $n,$ so that the probability of guessing at least $n$ correct answers is less than $\frac{1}{2},$ is :

Consider the following frequency distribution : <table class="pyq-table"><tbody><tr><td>class</td><td>$10-20$</td><td>$20-30$</td><td>$30-40$</td><td>$40-50$</td><td>$50-60$</td></tr><tr><td>Frequency</td><td>$\alpha$</td><td>$110$</td><td>$54$</td><td>$30$</td><td>$\beta$</td></tr></tbody></table>If the sum of all frequencies is $584$ and median is $45$, then $|\alpha -\beta |$ is equal to .

Let $X$ be a random variable with distribution. <table class="pyq-table"><tbody><tr><td>$x$</td><td>$-2$</td><td>$-1$</td><td>$3$</td><td>$4$</td><td>$6$</td></tr><tr><td>$P(X=x)$</td><td>$\frac{1}{5}$</td><td>$a$</td><td>$\frac{1}{3}$</td><td>$\frac{1}{5}$</td><td>$b$</td></tr></tbody></table>If the mean of $X$ is $2.3$ and variance of $X$ is ${\sigma }^{2},$ then $100{\sigma }^{2}$ is equal to :

The mean age of $25$ teachers in a school is $40$ years. A teacher retires at the age of $60$ years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is $39$ years, then the age (in years) of the newly appointed teacher is

A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is :

A seven digit number is formed using digits $3,3,4,4,4,5,5$. The probability, that number so formed is divisible by $2$, is

Consider the following frequency distribution: <table class="pyq-table"><tbody><tr><td>Class:</td><td>$0-6$</td><td>$6-12$</td><td>$12-18$</td><td>$18-24$</td><td>$24-30$</td></tr><tr><td>Frequency:</td><td>$a$</td><td>$b$</td><td>$12$</td><td>$9$</td><td>$5$</td></tr></tbody></table>If mean $=\frac{309}{22}$ and median $=14,$ then the value $(a-b{)}^{2}$ is equal to

The mean of $10$ numbers $7\times 8,10\times 10,13\times 12,16\times 14,\ldots$ is

Let ${X}_{1},{X}_{2},\ldots ,{X}_{18}$ be eighteen observations such that $\sum _{i=1}^{18}({X}_{i}-\alpha )=36$ and $\sum _{i=1}^{18}{({X}_{i}-\beta )}^{2}=90$, where $\alpha$ and $\beta$ are distinct real numbers. If the standard deviation of these observations is $1$, then the value of $|\alpha -\beta |$ is _______.

Words with or without meaning are to be formed using all the letters of the word $\mathrm{EXAMINATION}$. The probability that the letter $M$ appears at the fourth position in any such word is:

Consider the statistics of two sets of observations as follows: <table class="pyq-table"><tbody><tr><td></td><td>Size</td><td>Mean</td><td>Variance</td></tr><tr><td>Observation I</td><td>$10$</td><td>$2$</td><td>$2$</td></tr><tr><td>Observation II</td><td>$n$</td><td>$3$</td><td>$1$</td></tr></tbody></table>If the variance of the combined set of these two observations is $\frac{17}{9},$ then the value of $n$ is equal to ________.

Let $A,B$ and $C$ be three events such that the probability that exactly one of $A$ and $B$ occurs is $(1-k),$ the probability that exactly one of $B$ and $C$ occurs is $(1-2k),$ the probability that exactly one of $C$ and $A$ occurs is $(1-k)$ and the probability of all $A,B$ and $C$ occur simultaneously is ${k}^{2},$ where $0<k<1.$ Then the probability that at least one of $A,B$ and $C$ occur is:

Let $X$ be a random variable such that the probability function of a distribution is given by $P(X=0)=\frac{1}{2},P(X=j)=\frac{1}{{3}^{j}}(j=1,2,3,\ldots ,\infty ).$ Then the mean of the distribution and $P(X$ is positive and even$)$ respectively, are:

The coefficients $a,b$ and $c$ of the quadratic equation, $a{x}^{2}+bx+c=0$ are obtained by throwing a dice three times. The probability that this equation has equal roots is:

Let $S={1,2,3,4,5,6}.$ Then the probability that a randomly chosen onto function $g$ from $S$ to $S$ satisfies $g(3)=2g(1)$ is :

Let a computer program generate only the digits $0$ and $1$ to form a string of binary numbers with probability of occurrence of $0$ at even places be $\frac{1}{2}$ and probability of occurrence of $0$ at the odd place be $\frac{1}{3}.$ Then the probability that $10$ is followed by $01$ is equal to :

The probabilities of three events $A,B$ and $C$ are given $P(A)=0.6,P(B)=0.4$ and $P(C)=0.5$. If $P(A\cup B)=0.8,P(A\cap C)=0.3,P(A\cap B\cap C)=0.2,P(B\cap C)=\beta$ and $P(A\cup B\cup C)=\alpha$, where $0.85\leq \alpha \leq 0.95,$ then $\beta$ lies in the interval :

Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up a three or a five, is

The mean and variance of $8$ observations are $10$ and $13.5,$ respectively. If $6$ of these observations are $5,7,10,12,14,15,$ then the absolute difference of the remaining two observations is :