JEE Main 2022 — Probability & Statistics

A six faced die is biased such that $3\times P$(a prime number)$=6\times P$(a composite number)$=2\times P(1)$. Let $X$ be a random variable that counts the number of times one gets a perfect square on some throws of this die. If the die is thrown twice, then the mean of $X$ is

Bag $I$ contains $3$ red, $4$ black and $3$ white balls and Bag $\mathrm{II}$ contains $2$ red, $5$ black and $2$ white balls. One ball is transferred from Bag $I$ to Bag $\mathrm{II}$ and then a ball is draw from Bag $\mathrm{II}$. The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red, is

Bag $A$ contains $2$ white, $1$ black and $3$ red balls and bag $B$ contains $3$ black, $2$ red and $n$ white balls. One bag is chosen at random and $2$ balls drawn from it at random are found to be $1$ red and $1$ black. If the probability that both balls come from Bag $A$ is $\frac{6}{11}$, then $n$ is equal to _____

Five numbers ${x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5}$ are randomly selected from the numbers$1,2,3,\ldots \ldots ,18$ and are arranged in the increasing order $({x}_{1}<{x}_{2}<{x}_{1}<{x}_{4}<{x}_{2})$. The probability that ${x}_{2}=7$ and ${x}_{4}=11$ is

If a point $A(x,y)$ lies in the region bounded by the $y$-axis, straight lines $2y+x=6$ and $5x-6y=30$, then the probability that $y<1$ is

If $A$ and $B$ are two events such that $P(A)=\frac{1}{3},P(B)=\frac{1}{5}$ and $P(A\cup B)=\frac{1}{2}$, then $P(A{B}^{'})+P(B{A}^{'})$ is equal to

If the mean deviation about median for the number $3,5,7,2k,12,16,21,24$ arranged in the ascending order, is $6$ then the median is

If the mean deviation about the mean of the numbers $1,2,3,\ldots \ldots ,n$, where $n$ is odd, is $\frac{5(n+1)}{n}$, then $n$ is equal to ______.

If the numbers appeared on the two throws of a fair six faced die are $\alpha$ and $\beta$, then the probability that ${x}^{2}+\alpha x+\beta >0$, for all $x\in R$, is

If the probability that a randomly chosen $6$-digit number formed by using digits $1$ and $8$ only is a multiple of $21$ is $p$, then $96p$ is equal to _____.

In an examination, there are $10$ true-false type questions. Out of $10$, a student can guess the answer of $4$ questions correctly with probability $\frac{3}{4}$ and the remaining $6$ questions correctly with probability $\frac{1}{4}$. If the probability that the student guesses the answers of exactly $8$ questions correctly out of $10$ is $\frac{27k}{{4}^{10}}$, then $k$ is equal to

If P(A)=0.4, P(B)=0.5 and P(A∩B)=0.2 then P(A|B) is:

Let $A$ and $B$ be two events such that $P(B\mid A)=\frac{2}{5}$, $P(A\mid B)=\frac{1}{7}$ and $P(A\cap B)=\frac{1}{9}$. Consider $(S1)P({A}^{'}\cup B)=\frac{5}{6}$, $(S2)P({A}^{'}\cap {B}^{'})=\frac{1}{18}$. Then

Let ${E}_{1}$ and ${E}_{2}$ be two events such that the conditional probabilities $P({E}_{1}\mid {E}_{2})=\frac{1}{2}$, $P({E}_{2}\mid {E}_{1})=\frac{3}{4}$ and $P({E}_{1}\cap {E}_{2})=\frac{1}{8}$. Then

Let $S={E,{E}_{2}\ldots {E}_{8}}$ be a sample space of raddom experiment such that $P({E}_{n})=\frac{n}{36}$ for every $n=1,2\ldots .8$. Then the number of elements in the set ${A\subset S:P(A)\geq \frac{4}{5}}$ is _____.

Let $S$ be the sample space of all five digit numbers. If $p$ is the probability that a randomly selected number from $S$, is a multiple of $7$ but not divisible by $5$, then $9p$ is equal to

Let ${E}_{1},{E}_{2},{E}_{3}$ be three mutually exclusive events such that $P({E}_{1})=\frac{2+3p}{6},P({E}_{2})=\frac{2-p}{8}$ and $P({E}_{3})=\frac{1-p}{2}$. If the maximum and minimum values of $p$ are ${p}_{1}$ and ${p}_{2}$ then $({p}_{1}+{p}_{2})$ is equal to:

Let the mean and the variance of $20$ observations ${x}_{1},{x}_{2},\ldots {x}_{20}$ be $15$ and $9$, respectively. For $\alpha \in R$, if the mean of ${({x}_{1}+\alpha )}^{2},{({x}_{2}+\alpha )}^{2},\ldots ,{({x}_{20}+\alpha )}^{2}$ is $178$, then the square of the maximum value of $\alpha$ is equal to ______.

Let the mean and the variance of $5$ observations ${x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5}$ be $\frac{24}{5}$ and $\frac{194}{25}$ respectively. If the mean and variance of the first $4$ observation are $\frac{7}{2}$ and $a$ respectively, then $(4a+{x}_{5})$ is equal to

Let the mean of $50$ observations is $15$ and the standard deviation is $2$. However, one observation was wrongly recorded. The sum of the correct and incorrect observations is $70$. If the mean of the correct set of observations is $16$, then the variance of the correct set is equal to

Let $S={1,2,3,\ldots ,2022}$. Then the probability, that a randomly chosen number $n$ from the set $S$ such that $HCF(n,2022)=1$, is

Out of $60%$ female and $40%$ male candidates appearing in an exam, $60%$ candidates qualify it. The number of females qualifying the exam is twice the number of males qualifying it. A candidate is randomly chosen from the qualified candidates. The probability, that the chosen candidate is a female, is

Suppose a class has $7$ students. The average marks of these students in the mathematics examination is $62$, and their variance is $20$. A student fails in the examination if he/she gets less than $50$ marks, then in worst case, the number of students can fail is

The mean and standard deviation of $40$ observations are $30$ and $5$ respectively. It was noticed that two of these observations $12$ and $10$ were wrongly recorded. If $\sigma$ is the standard deviation of the data after omitting the two wrong observations from the data, then $38{\sigma }^{2}$ is equal to _______.

The mean and standard deviation of $15$ observations are found to be $8$ and $3$ respectively. On rechecking it was found that, in the observations, $20$ was misread as $5$. Then, the correct variance is equal to _____.

The mean and variance of $10$ observations were calculated as $15$ and $15$ respectively by a student who took by mistake $25$ instead of $15$ for one observation. Then, the correct standard deviation is _______.

The mean and variance of the data $4,5,6,6,7,8,x,y$ where $x<y$ are $6$ and $\frac{9}{4}$ respectively. Then ${x}^{4}+{y}^{2}$ is equal to

The mean of the numbers $a,b,8,5,10$ is $6$ and their variance is $6.8$. If $M$ is the mean deviation of the numbers about the mean, then $25M$ is equal to

The number of values of $a\in N$ such that the variance of $3,7,12,a,43-a$ is a natural number is:

The probability that a relation $R$ from ${x,y}$ to ${x,y}$ is both symmetric and transitive, is equal to:

The probability, that in a randomly selected $3$-digit number at least two digits are odd, is