JEE Main 2021 — Probability & Statistics

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 $9$ distinct balls be distributed among $4$ boxes, ${B}_{1},{B}_{2},{B}_{3}$ and ${B}_{4}$. If the probability that ${B}_{3}$ contains exactly $3$ balls is $k{(\frac{3}{4})}^{9}$ then $k$ lies in the set :

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 :

The probability that a randomly selected $2-$digit number belongs to the set ${n\in N:({2}^{n}-2)$ is a multiple of $3}$ is equal to

An online exam is attempted by $50$ candidates out of which $20$ are boys. The average marks obtained by boys is $12$ with a variance $2.$ The variance of marks obtained by $30$ girls is also $2.$ The average marks of all $50$ candidates is $15.$ If $\mu$ is the average marks of girls and ${\sigma }^{2}$ is the variance of marks of $50$ candidates, then $\mu +{\sigma }^{2}$ is equal to

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

Let the mean and variance of four numbers $3,7,x$ and $y(x>y)$ be $5$ and $10$ respectively. Then the mean of four numbers $3+2x,7+2y,x+y$ and $x-y$ is ______.

If the mean and variance of the following data: $6,10,7,13,a,12,b,12$ are $9$ and $\frac{37}{4}$ respectively, then ${(a-b)}^{2}$ is equal to:

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

Let in a series of $2n$ observations, half of them are equal to $a$ and remaining half are equal to $-a$. Also by adding a constant $b$ in each of these observations, the mean and standard deviation of new set become $5$ and $20$, respectively. Then the value of ${a}^{2}+{b}^{2}$ is equal to :

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?

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

If the variance of $10$ natural numbers $1,1,1,\ldots ,1,k$ is less than $10,$ then the maximum possible value of $k$ 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 _______.

Let ${B}_{i}(i=1,2,3)$ be three independent events in a sample space. The probability that only ${B}_{1}$ occur is $\alpha ,$ only ${B}_{2}$ occurs is $\beta$ and only ${B}_{3}$ occurs is $\gamma .$ Let $p$ be the probability that none of the events ${B}_{i}$ occurs and these $4$ probabilities satisfy the equations $(\alpha -2\beta )p=\alpha \beta$ and $(\beta -3\gamma )p=2\beta \gamma$ (All the probabilities are assumed to lie in the interval $(0,1))$ Then $\frac{P({B}_{1})}{P({B}_{3})}$ 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 the mean and variance of the frequency distribution <table class="pyq-table"><tbody><tr><td>$x:$</td><td>${x}_{1}=2$</td><td>${x}_{2}=6$</td><td>${x}_{3}=8$</td><td>${x}_{4}=9$</td></tr><tr><td>$f:$</td><td>$4$</td><td>$4$</td><td>$\alpha$</td><td>$\beta$</td></tr></tbody></table>be $6$ and $6.8$ respectively. If ${x}_{3}$ is changed from $8$ to $7,$ then the mean for the new data will be:

Let $n$ be an odd natural number such that the variance of $1,2,3,4,\ldots ,n$ is $14.$ Then $n$ is equal to ________.

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 ________.

If the mean and variance of six observations $7,10,11,15,a,b$ are $10$ and $\frac{20}{3},$ respectively, then the value of $|a-b|$ is equal to:

Two squares are chosen at random on a chessboard (see figure). The probability that they have a side in common is : 

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 :

When a certain biased die is rolled, a particular face occurs with probability $\frac{1}{6}-x$ and its opposite face occurs with probability $\frac{1}{6}+x.$ All other faces occur with probability $\frac{1}{6}.$ Note that opposite faces sum to $7$ in any die. If $0<x<\frac{1}{6},$ and the probability of obtaining total sum $=7,$ when such a die is rolled twice, is $\frac{13}{96},$ then the value of $x$ is

The probability distribution of random variable $X$ is given by: <table class="pyq-table"><tbody><tr><td>$X$</td><td>$1$</td><td>$2$</td><td>$3$</td><td>$4$</td><td>$5$</td></tr><tr><td>$P(X)$</td><td>$K$</td><td>$2K$</td><td>$2K$</td><td>$3K$</td><td>$K$</td></tr></tbody></table>Let $p=P(1<X<4\mid X<3)$. If $5p=\lambda K$, then $\lambda$ is equal to

A fair die is tossed until six is obtained on it. Let $X$ be the number of required tosses, then the conditional probability $P(X\geqslant 5\mid X>2)$ is :

Let $A$ and $B$ be independent events such that $P(A)=p,P(B)=2p$. The largest value of $p$, for which $P$ (exactly one of $A,B$ occurs)$=\frac{5}{9}$, 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:

Four dice are thrown simultaneously and the numbers shown on these dice are recorded in $2\times 2$ matrices. The probability that such formed matrices have all different entries and are non-singular, is:

Let there be three independent events ${E}_{1},{E}_{2}$ and ${E}_{3}.$ The probability that only ${E}_{1}$ occurs is $\alpha$ only ${E}_{2}$ occurs is $\beta$ and only ${E}_{3}$ occurs is $\gamma .$ Let $p''$ denote the probability of none of events occurs that satisfies the equations $(\alpha -2\beta )p=\alpha \beta$ and $(\beta -3\gamma )p=2\beta \gamma .$ All the given probabilities are assumed to lie in the interval $(0,1).$ Then, $\frac{\text{ Probability of occurrence of }{E}_{1}}{\text{ Probability of occurrence of }{E}_{3}}$ is equal to ________.

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:

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:

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

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 :

When a missile is fired from a ship, the probability that it is intercepted is $\frac{1}{3}$ and the probability that the missile hits the target, given that it is not intercepted, is $\frac{3}{4}.$ If three missiles are fired independently from the ship, then the probability that all three hit the target, is:

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:

An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number $2$ times is equal to the probability of getting an even number $3$ times, then the probability of getting an odd number for odd number of times is:

The probability that two randomly selected subsets of the set ${1,2,3,4,5}$ have exactly two elements in their intersection, 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 $A$ denote the event that a $6$-digit integer formed by $0,1,2,3,4,5,6$ without repetitions, be divisible by $3$ . Then probability of event $A$ is equal to :

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 :

A fair coin is tossed $n-$ times such that the probability of getting at least one head is at least $0.9.$ Then the minimum value of $n$ is _______.

The mean of $6$ distinct observations is $6.5$ and their variance is $10.25.$ If $4$ out of $6$ observations are $2,4,5$ and $7,$ then the remaining two observations are:

The mean and standard deviation of $20$ observations were calculated as $10$ and $2.5$ respectively. It was found that by mistake one data value was taken as $25$ instead of $35.$ If $\alpha$ and $\sqrt{\beta }$ are the mean and standard deviation respectively for correct data, then $(\alpha ,\beta )$ is:

The first of the two samples in a group has $100$ items with mean $15$ and standard deviation $3.$ If the whole group has $250$ items with mean $15.6$ and standard deviation $\sqrt{13.44},$ then the standard deviation of the second sample is:

An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first unit functions is $0.9$ and that of the second unit is $0.8.$ The instrument is switched on and it fails to operate. If the probability that only the first unit failed and second unit is functioning is $p,$ then $98p$ is equal to

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: