JEE Main Mathematics — Probability & Statistics previous year questions with solutions.
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 $A$ 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 :
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
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..........
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 $X={11,12,13,\ldots .,40,41}$ 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 $\mu$ 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 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 ____________.
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 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
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 ________ .
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
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 $S={M=[{a}_{ij}],{a}_{ij}\in {0,1,2},{1\leq i,j\leq 2}}$ be a sample space and $A{M\in S:M\text{is invertible}}$ be an even. Then $P(A)$ is equal to
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