JEE Main 2020 — Probability & Statistics

An unbiased coin is tossed $5$ times. Suppose that a variable $X$ is assigned the value $k$ when $k$ consecutive heads are obtained for $k=3,4,5,$ otherwise $X$ takes the value $-1.$ Then the expected value of $X,$ is

Box $1$ contains $30$ cards numbered $1$ to $30$ and Box $2$ contains $20$ cards numbered $31$ to $50$. A box is selected at random and a card is drawn from it. The number on the card is found to be a non-prime number. The probability that the card was drawn from Box $1$ is

Consider the data on $x$ taking the values $0,2,4,8,.....,{2}^{n}$ with frequencies $C0n,C1n,C2n,....,Cnn$ respectively. If the mean of this data is $\frac{728}{{2}^{n}}$, then $n$ is equal to ....... .

For the frequency distribution: Variate $(x):{x}_{1},{x}_{2},{x}_{3},\ldots ,{x}_{15}$ Frequency $(f):{f}_{1},{f}_{2},{f}_{3},\ldots ,{f}_{15}$ where $0<{x}_{1}<{x}_{2}<{x}_{3}<\ldots <{x}_{15}=10$ and $\sum _{i=1}^{15}{f}_{i}>0,$ the standard deviation cannot be

Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up a three or a five, is

If $\Sigma _{i=1}^{n}({x}_{i}-a)=n$ and $\Sigma _{i=1}^{n}{({x}_{i}-a)}^{2}=na,(n,a>1)$, then the standard deviation of $n$ observations ${x}_{1},{x}_{2},\ldots ,{x}_{n}$ is

If $10$ different balls are to be placed in $4$ distinct boxes at random, then the probability that two of these boxes contain exactly $2$ and $3$ balls is:

If the mean and the standard deviation of the data $3,5,7,a,b$ are $5$and $2$ respectively, then $a$ and $b$ are the roots of the equation:

If the mean and variance of eight numbers $3,7,9,12,13,20,x$ and $y$ be $10$ and $25$ respectively, then $x\cdot y$ is equal to

If the variance of the first $n$ natural numbers is $10$ and the variance of the first $m$ even natural numbers is $16,$ then the value of $m+n$ is equal to

If the variance of the following frequency distribution: Class: $10-2020-3030-40$ Frequency: $2x2$ is $50,$ then $x$ is equal to _______

If the variance of the terms in an increasing $A.P.$ ${b}_{1}{b}_{2},{b}_{3},\ldots \ldots ..,{b}_{11}$ is $90$ then the common difference of this $A.P.$ is

In a box, there are $20$ cards, out of which $10$ are labelled as $A$ and the remaining $10$ are labelled as $B$ . Cards are drawn at random, one after the other and with replacement, till a second $A$ card is obtained. The probability that the second $A$ card appears before the third $B$ card is:

In a game two players $A$ and $B$ take turns in throwing a pair of fair dice starting with player $A$ and total of scores on the two dice, in each throw is noted. $A$ wins the game if he throws a total of $6$ before $B$ throws a total of $7$ and $B$ wins the game if he throws a total of $7$ before $A$ throws a total of six. The game stops as soon as either of the players wins. The probability of $A$ winning the game is :

In a workshop, there are five machines and the probability of any one of them to be out of service on a day is $\frac{1}{4}$ . If the probability that at most two machines will be out of service on the same day is ${(\frac{3}{4})}^{3}k,$ then $k$ is equal to

A die is thrown 6 times. The probability of getting at least one six is:

Let $A$ and $B$, be two events such that the probability that exactly one of them occurs is $\frac{2}{5}$, and the probability that $A$ or $B$, occurs is $\frac{1}{2}$, then the probability of both of them occur together is.

Let $A$ and $B$ be two independent events such that $P(A)=\frac{1}{3}$ and $P(B)=\frac{1}{6}.$ Then, which of the following is true?

Let $X={x\in N:1\leq x\leq 17}$ and $Y={ax+b:x\in X\mathrm{and}a,b\in R,a>0}$. If mean and variance of elements of $Y$ are $17$ and $216$ respectively then $a+b$ is equal to

Let ${x}_{i}(1\leq i\leq 10)$ be ten observation of a random variable $X$. If $\sum _{i=1}^{10}({x}_{i}-p)=3$ and $\sum _{i=1}^{10}{({x}_{i}-p)}^{2}=9$ where $0\neq p\in R$, then the standard deviation of these observations is:

Let ${E}^{C}$ denote the complement of an event $E$. Let ${E}_{1},{E}_{2}$ and ${E}_{3}$ be any pairwise independent events with $P({E}_{1})>0$ and $P({E}_{1}\cap {E}_{2}\cap {E}_{3})=0$ then $P(({E}_{2}^{C}\cap {E}_{3}^{C})/{E}_{1})$ is equal to

Let the observation ${x}_{i}(1\leq i\leq 10)$ satisfy the equations $\sum _{i=1}^{10}({x}_{i}-5)=10$, $\sum _{i=1}^{10}{({x}_{i}-5)}^{2}=40$ . If $\mu$ and $\lambda$ are the mean and the variance of the observations, ${x}_{1}-3,{x}_{2}-3,....,{x}_{10}-3,$ then the ordered pair $(\mu ,\lambda )$ is equal to:

Out of 11 consecutive natural number if three numbers are selected at random (without repetition), then the probability that they are in A.P. with positive common difference is :

The mean and the standard deviation (s.d.) of $10$ observations are $20$ and $2$ respectively. Each of these $10$ observations is multiplied by $p$ and then reduced by $q,$ where $p\neq 0$ and $q\neq 0.$ If the new mean and new s.d. become half of their original values, then $q$ is equal to

The mean and variance of $8$ observations are $10$ and $13.5,$ respectively. If $6$ of these observations are $5,7,10,12,14,15,$ then the absolute difference of the remaining two observations is :

The mean and variance of $7$ observations are $8$ and $16$, respectively. If five observations are $2,4,10,12,14$ then the absolute difference of the remaining two observations is :

The mean and variance of $20$ observations are found to be $10$ and $4,$ respectively. On rechecking, it was found that an observation $9$ was incorrect and the correct observation was $11,$ then the correct variance is

The probabilities of three events $A,B$ and $C$ are given $P(A)=0.6,P(B)=0.4$ and $P(C)=0.5$. If $P(A\cup B)=0.8,P(A\cap C)=0.3,P(A\cap B\cap C)=0.2,P(B\cap C)=\beta$ and $P(A\cup B\cup C)=\alpha$, where $0.85\leq \alpha \leq 0.95,$ then $\beta$ lies in the interval :

The probability of a man hitting a target is $\frac{1}{10}.$ The least number of shots required, so that the probability of his hitting the target at least once is greater than $\frac{1}{4},$ is....

The probability that a randomly chosen $5$- digit number is made from exactly two digits is :