JEE Main 2023 — Probability & Statistics

The variance of the data 2, 4, 6, 8, 10 is:

The standard deviation of 5 observations 1, 2, 3, 4, 5 is:

Let $S={{w}_{1},{w}_{2},\ldots .}$ be the sample space associated to a random experiment. Let $P({w}_{n})=\frac{P({w}_{n-1})}{2},n\geq 2$. Let $A={2k+3l;k,l\in \mathbb{N}}$ and $B={{w}_{n};n\in A}$. Then $P(B)$ is equal to

A fair $n(n>1)$ faces die is rolled repeatedly until a number less than $n$ appears. If the mean of the number of tosses required is $\frac{n}{9}$, then $n$ is equal to

Three dice are rolled. If the probability of getting different numbers on the three dice is $\frac{p}{q},$ where $p$ and $q$ are co-prime, then $q-p$ is equal to

Let $S$ be the set of all values of ${a}_{1}$ for which the mean deviation about the mean of $100$ consecutive positive integers ${a}_{1},{a}_{2},{a}_{3},\ldots .,{a}_{100}$ is $25$ . Then $S$ is

Let $9={x}_{1}<{x}_{2}<\ldots <{x}_{7}$ be in an A.P. with common difference $d$. If the standard deviation of ${x}_{1},{x}_{2}\ldots ,{x}_{7}$ is 4 and the mean is $\bar{x}$, then $\bar{x}+{x}_{6}$ is equal to :

The mean and standard deviation of $10$ observations are $20$ and $8$ respectively. Later on, it was observed that one observation was recorded as $50$ instead of $40$. Then the correct variance is

If the variance of the frequency distribution <table class="pyq-table"><tbody><tr><td>${x}_{i}$</td><td>$2$</td><td>$3$</td><td>$4$</td><td>$5$</td><td>$6$</td><td>$7$</td><td>$8$</td></tr><tr><td>Frequency ${f}_{i}$</td><td>$3$</td><td>$6$</td><td>$16$</td><td>$\alpha$</td><td>$9$</td><td>$5$</td><td>$6$</td></tr></tbody></table>is $3$, then $\alpha$ is equal to

If the mean and variance of the frequency distribution <table class="pyq-table"><tbody><tr><td>${x}_{i}$</td><td>$2$</td><td>$4$</td><td>$6$</td><td>$8$</td><td>$10$</td><td>$12$</td><td>$14$</td><td>$16$</td></tr><tr><td>${f}_{i}$</td><td>$4$</td><td>$4$</td><td>$\alpha$</td><td>$15$</td><td>$8$</td><td>$\beta$</td><td>$4$</td><td>$5$</td></tr></tbody></table>are $9$ and $15.08$ respectively, then the value of ${\alpha }^{2}+{\beta }^{2}-\alpha \beta$ is _____.

The mean and variance of $5$ observations are $5$ and $8$ respectively. If $3$ observations are $1,3,5$, then the sum of cubes of the remaining two observations is

Let the mean of the data <table class="pyq-table"><tbody><tr><td>$x$</td><td>$1$</td><td>$3$</td><td>$5$</td><td>$7$</td><td>$9$</td></tr><tr><td>Frequency $(f)$</td><td>$4$</td><td>$24$</td><td>$28$</td><td>$\alpha$</td><td>$8$</td></tr></tbody></table>be $5.$ If $m$ and ${\sigma }^{2}$ are respectively the mean deviation about the mean and the variance of the data, then $\frac{3\alpha }{m+{\sigma }^{2}}$ is equal to $_______.$

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

Let $\Omega$ be the sample space and $A\subseteq \Omega$ be an event. Given below are two statements: (S1): If $P(A)=0$, then $A=\phi$ (S2): If $P(A)=$, then $A=\Omega$ Then

Let $A$ be the event that the absolute difference between two randomly chosen real numbers in the sample space $[0,60]$ is less than or equal to $a$. If $P(A)=\frac{11}{36}$, then a is equal to _____ .

If an unbiased die, marked with $-2,-1,0,1,2,3$ on its faces is thrown five times, then the probability that the product of the outcomes is positive, is :

Let the six numbers ${a}_{1},{a}_{2},...,{a}_{6}$ be in $A.P.$ and ${a}_{1}+{a}_{3}=10$.If the mean of these six numbers is $\frac{19}{2}$ and their variance is ${\sigma }^{2}$, then $8{\sigma }^{2}$ is equal to

Let M be the maximum value of the product of two positive integers when their sum is $66$ . Let the sample space $S={x\in Z:x(66-x)\geq \frac{5}{9}M}$ and the event $A={x\in S:x$ is a multiple of $3$}. Then $P(A)$ is equal to

Let the probability of getting head for a biased coin be $\frac{1}{4}$. It is tossed repeatedly until a head appears. Let $N$ be the number of tosses required. If the probability that the equation $64{x}^{2}+5Nx+1=0$ has no real root is $\frac{p}{q}$, where $p$ and $q$ are co-prime, then $q-p$ is equal to..........

Let $\mu$ be the mean and $\sigma$ be the standard deviation of the distribution <table class="pyq-table"><tbody><tr><td>${X}_{i}$</td><td>$0$</td><td>$1$</td><td>$2$</td><td>$3$</td><td>$4$</td><td>$5$</td></tr><tr><td>${f}_{i}$</td><td>$k+2$</td><td>$2k$</td><td>${k}^{2}-1$</td><td>${k}^{2}-1$</td><td>${k}^{2}+1$</td><td>$k-3$</td></tr></tbody></table>where $\Sigma {f}_{i}=62$. If $[x]$ denotes the greatest integer $\leq x$, then$[{\mu }^{2}+{\sigma }^{2}]$ is equal to

Let $X={11,12,13,\ldots .,40,41}$ and $Y={61,62,63,...,90,91}$ be the two sets of observations. If $\bar{x}$ and $\bar{y}$ are their respective means and ${\sigma }^{2}$ is the variance of all the observations in $X\cup Y$, then $|\bar{x}+\bar{y}-{\sigma }^{2}|$ is equal to ________

$25%$ of the population are smokers. A smoker has $27$ times more chances to develop lung cancer then a non-smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is $\frac{k}{10}$.Then the value of $k$ is _____ .

In a bolt factory, machines $A,B$ and $C$ manufacture respectively $20%,30%$ and $50%$ of the total bolts. Of their output $3,4$ and $2$ percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found the defective then the probability that it is manufactured by the machine $C$ is

Let $S={M=[{a}_{ij}],{a}_{ij}\in {0,1,2},{1\leq i,j\leq 2}}$ be a sample space and $A{M\in S:M\text{is invertible}}$ be an even. Then $P(A)$ is equal to

The mean and variance of $7$ observations are $8$ and $16$ respectively. If one observation $14$ is omitted, $a$ and $b$ are respectively mean and variance of remaining $6$ observation, then $a+3b-5$ is equal to ________

Let the mean and standard deviation of marks of class A of $100$ students be respectively $40$ and $\alpha (>0)$, and the mean and standard deviation of marks of class $B$ of $n$ students be respectively $55$ and $30-\alpha$. If the mean and variance of the marks of the combined class of $100+n$ students are respectively $50$ and $350$ , then the sum of variances of classes $A$ and $B$ is

A bag contains $6$ white and $4$ black balls. A die is rolled once and the number of balls equal to the number obtained on the die are drawn from the bag at random. The probability that all the balls drawn are white is

Let the mean and variance of $12$ observations be $\frac{9}{2}$ and $4$ respectively. Later on, it was observed that two observations were considered as $9$ and $10$ instead of $7$ and $14$ respectively. If the correct variance is $\frac{m}{n}$, where $m$ and $n$ are coprime, then $m+n$ are coprime, then $m+n$ is equal to

A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is $p$. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colours is $q$. If $p:q=m$ $:n$, where $m$ and $n$ are co-prime, then $m+n$ is equal to

The mean and variance of a set of $15$ numbers are $12$ and $14$ respectively. The mean and variance of another set of $15$ numbers are $14$ and ${\sigma }^{2}$ respectively. If the variance of all the $30$ numbers in the two sets is $13$, then ${\sigma }^{2}$ is equal to

If the mean of the frequency distribution <table class="pyq-table"><tbody><tr><td>Class :</td><td>$0-10$</td><td>$10-20$</td><td>$20-30$</td><td>$30-40$</td><td>$40-50$</td></tr><tr><td>Frequency :</td><td>$2$</td><td>$3$</td><td>$x$</td><td>$5$</td><td>$4$</td></tr></tbody></table>is $28$, then its variance is ________ .

Let sets $A$ and $B$ have $5$ elements each. Let the mean of the elements in sets $A$ and $B$ be $5$ and $8$ respectively and the variance of the elements in sets $A$ and $B$ be $12$ and $20$ respectively. A new set $C$ of $10$ elements is formed by subtracting $3$ from each element of $A$ and adding $2$ to each element of $B$. Then the sum of the mean and variance of the elements of $C$ is

The mean and variance of the marks obtained by the students in a test are $10$ and $4$ respectively. Later, the marks of one of the students is increased from $8$ to $12$ . If the new mean of the marks is $10.2$. then their new variance is equal to:

A coin is biased so that the head is $3$ times as likely to occur as tail. This coin is tossed until a head or three tails occur. If $X$ denotes the number of tosses of the coin, then the mean of $X$ is

Two dice are thrown independently. Let $A$ be the event that the number appeared on the ${1}^{\text{st }}$ die is less than the number appeared on the ${2}^{\text{nd }}$ die, $B$ be the event that the number appeared on the ${1}^{\text{st }}$ die is even and that on the second die is odd, and $C$ be the event that the number appeared on the ${1}^{\text{st }}$ die is odd and that on the ${2}^{\text{nd }}$ is even. Then

Let the mean of 6 observations $1,2,4,5,x$ and $y$ be $5$ and their variance be $10$. Then their mean deviation about the mean is equal to

Let $N$ denote the sum of the numbers obtained when two dice are rolled. If the probability that ${2}^{N}<N!$ is $\frac{m}{n}$ where $m$ and $n$ are coprime, then $4m-3n$ is equal to

The urns $A,B$ and $C$ contains $4$ red, $6$ black; $5$ red, $5$ black and $\lambda$ red, $4$ black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball drawn is red and the probability that it is drawn from urn $C$ is $0.4$, then the square of length of the side of largest equilateral triangle, inscribed in the parabola ${y}^{2}=\lambda x$ with one vertex at vertex of parabola is

Let the positive numbers ${a}_{1},{a}_{2,}{a}_{3},{a}_{4}$ and ${a}_{5}$ be in a G.P. Let their mean and variance be $\frac{31}{10}$ and $\frac{m}{n}$ respectively, where $m$ and $n$ are co-prime. If the mean of their reciprocals is $\frac{31}{10}$ and ${a}_{3}+{a}_{4}+{a}_{5}=14$, then $m+n$ is equal to ____________.

There rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable X denote the number of rotten apples. If $\mu$ and ${\sigma }^{2}$ represent mean and variance of X, respectively, then $10({\mu }^{2}+{\sigma }^{2})$ is equal to

The mean and standard deviation of the marks of $10$ students were found to be $50$ and $12$ respectively. Later, it was observed that two marks $20$ and $25$ were wrongly read as $45$ and $50$ respectively. Then the correct variance is

Let the mean and variance of $8$ numbers $x,y,10,12,6,12,4,8$ be $9$ and $9.25$ respectively. If $x>y,$ then $3x-2y$ is equal to $_______$

Let $N$ be the sum of the numbers appeared when two fair dice are rolled and let the probability that $N-2,\sqrt{3N},N+2$ are in geometric progression be $\frac{k}{48}$. Then the value of $k$ is

Two dice $A$ and $B$ are rolled. Let the numbers obtained on $A$ and $B$ be $\alpha$ and $\beta$ respectively. If the variance of $\alpha -\beta$ is $\frac{p}{q},$ where $p$ and $q$ are co-prime, then the sum of the positive divisors of $p$ is equal to