JEE Main Mathematics — Probability & Statistics previous year questions with solutions.
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:
A fair die is thrown until $2$ appears. Then the probability, that $2$ appears in even number of throws, is
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
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 _________
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 :
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
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
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 ______
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 :
The variance of the data 2, 4, 6, 8, 10 is:
The standard deviation of 5 observations 1, 2, 3, 4, 5 is:
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
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 $S={{w}_{1},{w}_{2},\ldots .}$ 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
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 $_______.$
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 :
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
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
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 ________
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 _____.