JEE Main 2024 — Probability & Statistics

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 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

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 :

Bag$A$ contains $3$ white, $7$ red balls and bag $B$ contains $3$ white, $2$ red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from the bag $A$, if the ball drawn in white, 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:

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 ______

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 _________

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

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

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

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

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}

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

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:

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 ______

A bag contains 5 red and 3 blue balls. Two balls are drawn at random without replacement. The probability that both are red is:

In a binomial distribution with n=5 and p=1/3, the probability of exactly 2 successes 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 $\mathrm{a}, \mathrm{b}$ 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 ________

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__________

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 :

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 :

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 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

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:

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

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 :

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 :

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 :

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

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 _________.

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 :

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________

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

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 :

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 :

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