JEE Main 2024 — Probability & Statistics

Let $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked $1,2,3,4$. If the probability that $a x^2+b x+c=0$ has all real roots is $\frac{m}{n}$, $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then $\mathrm{m}+\mathrm{n}$ is equal to ________

There are three bags $X, Y$ and $Z$. Bag $X$ contains 5 one-rupee coins and 4 five-rupee coins; Bag $Y$ contains 4 one-rupee coins and 5 five-rupee coins and Bag Z contains 3 one-rupee coins and 6 five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability, that it came from bag Y, is :

From a lot of 10 items, which include 3 defective items, a sample of 5 items is drawn at random. Let the random variable $\mathrm{X}$ denote the number of defective items in the sample. If the variance of $\mathrm{X}$ is $\sigma^2$, then $96 \sigma^2$ is equal to ______

The frequency distribution of the age of students in a class of 40 students is given below. $\begin{array}{|l|c|c|c|c|l|l|} \hline \text{Age} & 15 & 16 & 17 & 18 & 19 & 20 \\ \hline \text{No of Students} & 5 & 8 & 5 & 12 & x & y \\ \hline \end{array}$ If the mean deviation about the median is 1.25, then $4 x+5 y$ is equal to :

If the mean of the following probability distribution of a random variable $\mathrm{X}$ : $\begin{array}{|c|c|c|c|c|c|} \hline \mathrm{X} & 0 & 2 & 4 & 6 & 8 \\ \hline \mathrm{P}(\mathrm{X}) & a & 2 a & a+b & 2 b & 3 b \\ \hline \end{array}$ is $\frac{46}{9}$, then the variance of the distribution is

Let the median and the mean deviation about the median of $7$ observation $170,125,230,190,210,a,b$ be $170$ and $\frac{205}{7}$ respectively. Then the mean deviation about the mean of these $7$ observations is:

If the variance ${\sigma }^{2}$ of the data $\begin{matrix}{x}_{i} & 0 & 1 & 5 & 6 & 10 & 12 & 17 \\ {f}_{i} & 3 & 2 & 3 & 2 & 6 & 3 & 3\end{matrix}$ is $k$ then the value of $[k]$ is ______ {where $[.]$ denotes the greatest integer funciton}

The mean and standard deviation of $15$ observations were found to be $12$ and $3$ respectively. On rechecking it was found that an observation was read as $10$ in place of $12.$ If $\mu$ and ${\sigma }^{2}$ denote the mean and variance of the correct observations respectively, then $15(\mu +{\mu }^{2}+{\sigma }^{2})$ is equal to _________.

Let ${a}_{1},{a}_{2},...,{a}_{10}$ be $10$ observations such that $\sum _{k=1}^{10}{a}_{k}=50$ and $\underset{\forall k<j}{\sum }{a}_{k}\cdot {a}_{j}=1100$. Then the standard deviation of ${a}_{1},{a}_{2},\ldots ,{a}_{10}$ is equal to :

If the variance of the frequency distribution $\begin{array}{|c|c|c|c|c|c|c|} \hline x & c & 2 c & 3 c & 4 c & 5 c & 6 c \\ \hline f & 2 & 1 & 1 & 1 & 1 & 1 \\ \hline \end{array}$ is 160, then the value of $c \in N$ is

Let Ajay will not appear in JEE exam with probability $p=\frac{2}{7},$ while both Ajay and Vijay will appear in the exam with probability $q=\frac{1}{5}$. Then the probability, that Ajay will appear in the exam and Vijay will not appear is:

Let $M$ denote the median of the following frequency distribution. $\begin{matrix}\text{ Class } & 0-4 & 4-8 & 8-12 & 12-16 & 16-20 \\ \text{ Frequency } & 3 & 9 & 10 & 8 & 6\end{matrix}$ Then $20M$ is equal to :

The coefficients $a, b, c$ in the quadratic equation $a x^2+b x+c=0$ are from the set $\{1,2,3,4,5,6\}$. If the probability of this equation having one real root bigger than the other is $p$, then 216 p equals :

If the mean and variance of five observations are $\frac{24}{5}$ and $\frac{194}{25}$ respectively and the mean of first four observations is $\frac{7}{2}$, then the variance of the first four observations in equal to

Let $\alpha, \beta \in \mathbf{R}$. Let the mean and the variance of 6 observations $-3,4,7,-6, \alpha, \beta$ be 2 and 23 , respectively. The mean deviation about the mean of these 6 observations is :

Let $\mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathrm{N}$ and $\mathrm{a} < \mathrm{b} < \mathrm{c}$. Let the mean, the mean deviation about the mean and the variance of the 5 observations $9,25, \mathrm{a}, \mathrm{b}, \mathrm{c}$ be 18,4 and $\frac{136}{5}$, respectively. Then $2 \mathrm{a}+\mathrm{b}-\mathrm{c}$ is equal to__________

The coefficients $a, b, c$ in the quadratic equation $a x^2+b x+c=0$ are chosen from the set $\{1,2,3,4,5,6,7,8\}$. The probability of this equation having repeated roots is :

Let the mean and the standard deviation of the probability distribution $\begin{array}{|c|c|c|c|c|} \hline \mathrm{X} & \mathrm{\alpha} & \mathrm{1} & 0 & -3 \\ \hline \mathrm{P}(\mathrm{X}) & \frac{1}{3} & \mathrm{~K} & \frac{1}{6} & \frac{1}{4} \\ \hline \end{array}$ be $\mu$ and $\sigma$, respectively. If $\sigma-\mu=2$, then $\sigma+\mu$ is equal to________

Let the mean and the variance of $6$ observation $a,b,68,44,48,60$ be $55$ and $194$, respectively if $a>b$, then $a+3b$ is

Consider $10$ observation ${x}_{1},{x}_{2},...{x}_{10}$, such that $\sum _{i=1}^{10}({x}_{i}-\alpha )=2$ and $\sum _{i=1}^{10}{({x}_{i}-\beta )}^{2}=40$, where $\alpha ,\beta$ are positive integers. Let the mean and the variance of the observations be $\frac{6}{5}$ and $\frac{84}{25}$ respectively. The $\frac{\beta }{\alpha }$ is equal to:

Two marbles are drawn in succession from a box containing $10$ red, $30$ white, $20$ blue and $15$ orange marbles, with replacement being made after each drawing. Then the probability, that first drawn marble is red and second drawn marble is white, is

Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables $X$ and $Y$ respectively denote the number of blue and yellow balls. If $\bar{X}$ and $\bar{Y}$ are the means of $X$ and $Y$ respectively, then $7 \bar{X}+4 \bar{Y}$ is equal to________

If an unbiased dice is rolled thrice, then the probability of getting a greater number in the $i^{\text {th }}$ roll than the number obtained in the $(i-1)^{\text {th }}$ roll, $i=2,3$, is equal to

Let the sum of two positive integers be 24 . If the probability, that their product is not less than $\frac{3}{4}$ times their greatest possible product, is $\frac{m}{n}$, where $\operatorname{gcd}(m, n)=1$, then $n-m$ equals

A company has two plants $A$ and $B$ to manufacture motorcycles. $60 \%$ motorcycles are manufactured at plant $A$ and the remaining are manufactured at plant $B .80 \%$ of the motorcycles manufactured at plant $A$ are rated of the standard quality, while $90 \%$ of the motorcycles manufactured at plant $B$ are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If $p$ is the probability that it was manufactured at plant $B$, then $126 p$ is

From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random variable $\mathrm{X}$ denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If variance of $X$ is $\frac{m}{n}$, where $\operatorname{gcd}(m, n)=1$, then $n-m$ is equal to _________

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as $\frac{1}{3}$ and $\frac{2}{3}$ respectively. Let $x$ be the number of matches that the team wins, and $y$ be the number of matches that team loses. If the probability $\mathrm{P}(|x-y| \leq$ 2) is $p$, then $3^9 p$ equals ______

Three rotten apples are accidently mixed with fifteen good apples. Assuming the random variable $x$ to be the number of rotten apples in a draw of two apples, the variance of $x$ is

A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed $3$ times, then the probability of getting two tails and one head is-

An integer is chosen at random from the integers $1,2,3,.....,50$. The probability that the chosen integer is a multiple of atleast one of $4,6$ and $7$ is

Two integers $x\text{and}y$ are chosen with replacement from the set ${0,1,2,3,\ldots ..,10}$. Then the probability that $|x-y|>5$ is :

A fair die is thrown until $2$ appears. Then the probability, that $2$ appears in even number of throws, is

A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required and let $a=P(X=3),b=P(X\geq 3)$ and $c=$ $P(X\geq 6\mid X>3)$. Then $\frac{b+c}{a}$ is equal to

An urn contains $6$ white and $9$ black balls. Two successive draws of $4$ balls are made without replacement. The probability, that the first draw gives all white balls and the second draw gives all black balls, is :

A bag contains $8$ balls, whose colours are either white or black. $4$ balls are drawn at random without replacement and it was found that $2$ balls are white and other $2$ balls are black. The probability that the bag contains equal number of white and black balls is:

If the mean and variance of the data $65,68,58,44$, $48,45,60,\alpha ,\beta ,60$ where $\alpha >\beta$ are $56$ and $66.2$ respectively, then ${\alpha }^{2}+{\beta }^{2}$ is equal to

Three urns A, B and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn $\mathrm{A}$ is :

If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is:

The mean and standard deviation of 20 observations are found to be 10 and 2 . respectively. On rechecking, it was found that an observation by mistake was taken 8 instead of 12. The correct standard deviation is