Probability & Statistics PYQ — Page 15

Let $A$ and $B$ be two events such that $P(\bar{A\cup B})=\frac{1}{6},P(A\cap B)=\frac{1}{4}$ and $P(\bar{A})=\frac{1}{4},$ where $\bar{A}$ stands for the complement of the event $A$. Then the events $A$ and $B$ are

In a set of $2n$ distinct observations, each of the observation below the median of all the observations is increased by $5$ and each of the remaining observations is decreased by $3$. Then, the mean of the new set of observations :

Let $\bar{x}$, $M$ and ${\sigma }^{2}$ be respectively the mean, mode and variance of $n$ observations ${x}_{1},{x}_{2},....,{x}_{n}$ and ${d}_{i}=-{x}_{i}-a,i=1,2,....,n,$ where $a$ is any number. Statement I: Variance of ${d}_{1},{d}_{2},...,{d}_{n}$ is ${\sigma }^{2}$. Statement II: Mean and mode of ${d}_{1},{d}_{2},....,{d}_{n}$ are $-\bar{x}-a$ and $-M-a$, respectively.

Given two independent events, if the probability that exactly one of them occurs is $\frac{26}{49}$ and the probability that none of them occurs is $\frac{15}{49}$, then the probability of more probable of the two events is :

If the events $\mathrm{A}$ and $\mathrm{B}$ are mutually exclusive events such that $\mathrm{P}(\mathrm{A})=\frac{3 x+1}{3}$ and $\mathrm{P}(\mathrm{B})=\frac{1-x}{4}$, then the set of possible values of $x$ lies in the interval :

Mean of 5 observations is 7 . If four of these observations are $6,7,8,10$ and one is missing then the variance of all the five observations is :

$\mathrm{A}, \mathrm{B}, \mathrm{C}$ try to hit a target simultaneously but independently. Their respective probabilities of hitting the targets are $\frac{3}{4}, \frac{1}{2}, \frac{5}{8}$. The probability that the target is hit by A or B but not by $\mathrm{C}$ is :

The mean of a data set consisting of 20 observations is 40 . If one observation 53 was wrongly recorded as 33 , then the correct mean will be:

If the median and the range of four numbers $\{x, y, 2 x+y, x-y\}$, where $0 < y < x < 2 y$, are 10 and 28 respectively, then the mean of the numbers is :

The probability of a man hitting a target is $\frac{2}{5}$. He fires at the target $k$ times $(k$, a given number). Then the minimum $k$, so that the probability of hitting the target at least once is more than $\frac{7}{10}$, is :

In a set of $2 n$ observations, half of them are equal to ' $\mathrm{a}$ ' and the remaining hall are equal to ' $-\mathrm{a}^{\prime}$ '. If the standard deviation of all the observations is 2 ; then the value of $|a|$ is :

All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of $10$ to each of the students. Which of the following statistical measures will not change even after the grace marks were given ?

The median of 100 observations grouped in classes of equal width is 25 . If the median class interval is 20-30 and the number of observations less than 20 is 45 , then the frequency of median class is

Let $\mathrm{x}_1, \mathrm{x}_2, \ldots \ldots, \mathrm{x}_{\mathrm{n}}$ be $\mathrm{n}$ observations, and let $\overline{\mathrm{x}}$ be their arithematic mean and $\sigma^2$ be their variance. Statement $1$: Variance of $2 x_1, 2 x_2, \ldots \ldots, 2 x_n$ is $4 \sigma^2$. Statement $2$: Arithmetic mean of $2 \mathrm{x}_1, 2 \mathrm{x}_2, \ldots . ., 2 \mathrm{x}_{\mathrm{n}}$ is $4 \overline{\mathrm{x}}$.

A number $n$ is randomly selected from the set $\{1,2,3, \ldots ., 1000\}$. The probability that $\frac{\sum_{i=1}^n i^2}{\sum_{i=1}^n i}$ is an integer is

The frequency distribution of daily working expenditure of families in a locality is as follows:  If the mode of the distribution is $₹ 140$, then the value of $b$ is

If six students, including two particular students $A$ and $B$, stand in a row, then the probability that $A$ and $B$ are separated with one student in between them is

There are two balls in an urn. Each ball can be either white or black. If a white ball is put into the urn and there after a ball is drawn at random from the urn, then the probability that it is white is

Statement 1: The variance of first $n$ odd natural numbers is $\frac{n^2-1}{3}$ Statement 2: The sum of first $\mathrm{n}$ odd natural number is $n^2$ and the sum of square of first $n$ odd natural numbers is $\frac{n\left(4 n^2+1\right)}{3}$.

Three numbers are chosen at random without replacement from $\{1,2,3, \ldots . .8\}$. The probability that their minimum is $3$ , given that their maximum is $6$ , is

If the mean of $4,7,2,8,6$ and a is 7 , then the mean deviation from the median of these observations is

If $C$ and $D$ are two events such that $C \subset D$ and $P(D) \neq 0$, then the correct statement among the following is

If the mean deviation about the median of the numbers $\mathrm{a}, 2 \mathrm{a}, \ldots, 50 \mathrm{a}$ is 50 , then $|\mathrm{a}|$ equals

For two data sets, each of size 5 , the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4 , respectively. The variance of the combined data set is