JEE Main 2019 — Probability & Statistics

If the standard deviation of the numbers $-1,0,1,k$ is $\sqrt{5}$ where $k>0,$ then $k$ is equal to

Four persons can hit a target correctly with probabilities $\frac{1}{2},\frac{1}{3},\frac{1}{4}$ and $\frac{1}{8}$ respectively. If all hit at the target independently, then the probability that the target would be hit, is

If for some $x\in R$, the frequency distribution of the marks obtained by $20$ students in a test is: <table class="pyq-table"><tbody><tr><td>Marks</td><td>2</td><td>3</td><td>5</td><td>7</td></tr><tr><td>Frequency distribution</td><td>${(x+1)}^{2}$</td><td>$(2x-5)$</td><td>${x}^{2}-3x$</td><td>$x$</td></tr></tbody></table>Then the mean of the marks is :

Let $A$ and $B$ be two non-null events such that $A\subset B.$ Then, which of the following statements is always correct?

The outcome of each of 30 items was observed; 10 items gave an outcome $\frac{1}{2}-\mathrm{d}$ each, 10 items gave outcome $\frac{1}{2}$ each and the remaining 10 items gave outcome $\frac{1}{2}+\mathrm{d}$ each. If the variance of this outcome data is $\frac{4}{3}$ then $|\mathrm{d}|$ equals:

The minimum number of times one has to toss a fair coin so that the probability of observing at least one head is at least $90%$ is:

$5$ students of a class have an average height $150 cm$ and variance $18 c{m}^{2}.$ A new student, whose height is $156 cm,$ joined them. The variance $(in c{m}^{2})$ of the height of these six students is:

If both the mean and the standard deviation of $50$ observations ${x}_{1}, {x}_{2}, \ldots , {x}_{50}$ are equal to $16$, then the mean of ${({x}_{1}-4)}^{2}, {({x}_{2}-4)}^{2}, \ldots ,{({x}_{50}-4)}^{2}$ is

The mean and the median of the following ten numbers in increasing order $10, 22, 26, 29, 34, x, 42, 67, 70, y$ are $42$ and $35$ respectively, then $\frac{y}{x}$ is equal to:

Two newspapers $A$ and $B$ are published in a city. It is known that $25%$ of the city population reads $A$ and $20%$ reads $B$ while $8%$ reads both $A$ and $B.$ Further, $30%$ of those who read $A$ but not $B$ look into advertisements and $40%$ of those who read $B$ but not $A$ also look into advertisements, while $50%$ of those who read both $A$ and $B$ look into advertisements. Then the percentage of the population who look into advertisements is:

In a class of $60$ students, $40$ opted for NCC, $30$ opted for NSS and $20$ opted for both NCC and NSS. If one of these students is selected at random, then the probability that the student selected has opted neither for NCC nor for NSS is :

An urn contains $5$ red and $2$ green balls. A ball is drawn at random from the urn. If the drawn ball is green, then a red ball is added to the urn and if the drawn ball is red, then a green ball is added to the urn; the original ball is not returned to the urn. Now, a second ball is drawn at random from it. The probability that the second ball is red, is:

A data consists of $n$ observations: ${x}_{1}, {x}_{2},\ldots , {x}_{n}.$ If $\sum _{i=1}^{n}{({x}_{i}+1)}^{2}=9n$ and $\sum _{i=1}^{n}{({x}_{i}-1)}^{2}=5n$, then the standard deviation of this data is

A bag contains 30 white balls and 10 red balls. 16 balls are drawn one by one randomly from the bag with replacement. If $X$ be the number of white balls drawn, then $\left(\frac{\text { mean of } \mathrm{X}}{\text { standard deviation of } \mathrm{X}}\right)$ is equal to:

In a game, a man wins Rs. $100$ if he gets $5$ or $6$ on a throw of a fair die and loses Rs. $50$ for getting any other number on the die. If he decides to throw the die either till he gets a five or a six or to a maximum of three throws, then his expected gain/loss (in rupees) is :

The mean and the variance of five observations are $4$ and $5.20$, respectively. If three of the observations are $3,4$ and $4$; then the absolute value of the difference of the other two observations, is :

In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to :

The mean of five observations is $5$ and their variance is $9.20.$ If three of the given five observations are $1,3$ and $8,$ then a ratio of other two observations is

A person throws two fair dice. He wins Rs. $15$ for throwing a doublet (same numbers on the two dice), wins Rs $12$ when the throw results in the sum of $9$ , and loses Rs. $6$ for any other outcome on the throw. Then the expected gain/loss (in Rs.) of the person is:

Two integers are selected at random from the set $\{1,2, \ldots, 11\}$. Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is :

If the data ${x}_{1},{x}_{2},\ldots {x}_{10}$ is such that the mean of first four of these is $11,$ the mean of the remaining six is $16$ and the sum of squares of all of these is $2000$, then the standard deviation of this data is:

If the mean and standard deviation of $5$ observations ${x}_{1},{x}_{2}, {x}_{3}, {x}_{4}, {x}_{5}$ are $10$ and $3,$ respectively, then the variance of $6$ observations ${x}_{1}, {x}_{2}, \ldots , {x}_{5}$ and $-50$ is equal to

Two cards are drawn successively with replacement from a well-shuffled deck of $52$ cards. Let $X$ denote the random variable of number of aces obtained in the two drawn cards. Then $P(X=1)+P(X=2)$ equals:

If the sum of the deviations of $50$ observations from $30$ is $50$, then the mean of these observations is :

If the probability of hitting a target by a shooter, in any shot is $\frac{1}{3},$ then the minimum number of independent shots at the target required by him so that the probability of hitting the target at least once is greater than $\frac{5}{6},$ is:

A student scores the following marks in five tests: $45,54,41,57,43$. His score is not known for the sixth test. If the mean score is $48$ in the six tests, then the standard deviation of the marks in six tests is:

Assume that each born child is equally likely to be a boy or girl. If two families have two children each, then the conditional probability that all children are girls given that at least two are girls is:

The mean and variance for seven observations are $8$ and $16$ respectively. If $5$ of the observations are $2,4,10,12,14,$ then the product of the remaining two observations is

An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well-shuffled pack of nine cards numbered $1,2,3,\ldots ,9$ is randomly picked and the number on the card is noted. The probability that the noted number is either $7$ or $8$ is

Let $\mathrm{S}=\{1,2, \ldots . ., 20\}$. A subset $\mathrm{B}$ of $\mathrm{S}$ is said to be "nice", if the sum of the elements of $\mathrm{B}$ is 203 . Than the probability that a randomly chosen subset of $S$ is "nice" is :