Probability & Statistics PYQ — Page 14

For three events, $A, B$ and $C,$ $P$(Exactly one of $A$ or $B$ occurs) $=P$(Exactly one of $B$ or $C$ occurs) $=P$(Exactly one of $C$ or $A$ occurs) $=\frac{1}{4}$ and $P$(All the three events occur simultaneously) $=\frac{1}{16}.$ Then the probability that at least one of the events occurs, is:

The sum of $100$ observations and the sum of their squares are $400&2475$, respectively. Later on, three observations $3,4&5$ were found to be incorrect. If the incorrect observations are omitted, then the variance of the remaining observations is

The mean age of $25$ teachers in a school is $40$ years. A teacher retires at the age of $60$ years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is $39$ years, then the age (in years) of the newly appointed teacher is

Let $E&F$ be two independent events. The probability that $E&F$ happen is $\frac{1}{12}$ and the probability that neither $E$ nor $F$ happens is $\frac{1}{2}$ , then a value of $\frac{P(E)}{P(F)}$ is:

A box contains $15$ green and $10$ yellow balls. If $10$ balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is:

From a group of $10$ men and $5$ women, four member committees are to be formed each of which must contain at least one women. Then the probability for these committees to have more women than men, is :

If $A$ and $B$ are any two events such that $P(A)=\frac{2}{5}$ and $P(A\cap B)=\frac{3}{20},$ then the conditional probability, $P(A|({A}^{'}\cup {B}^{'})),$ where ${A}^{'}$ denotes the complement of $A$, is equal to :

If the mean deviation of the numbers $1, 1+d,\ldots , 1+100d$ from their mean is $255$, then a value of $d$ is :

Let two fair six-faced dice $A$ and $B$ be thrown simultaneously. If ${E}_{1}$ is the event that die $A$ shows up four, ${E}_{2}$ is the event that die $B$ shows up two and ${E}_{3}$ is the event that the sum of numbers on both dice is odd, then which of the following statements is not true?

If the standard deviation of the numbers $2,3,a$ and $11$ is $3.5$, then which of the following is true ?

The mean of $5$ observations is $5$ and their variance is $12.4$. If three of the observations are $1,2&6$; then the value of the remaining two is :

Let $X$ be a set containing $10$ elements and $P(X)$ be its power set. If $A$ and $B$ are picked up at random from $P(X)$, with replacement, then the probability that $A$ and $B$ have equal number of elements is:

If the mean and the variance of a binomial variate $X$ are $2&1$ respectively, then the probability that $X$ takes a value greater than or equal to one is:

If the lengths of the sides of a triangle are decided by the three throws of a single fair die, then the probability that the triangle is of maximum area given that it is an isosceles triangle, is:

The mean of a data set comprising of $16$ observations is $16$. If one of the observation value $16$ is deleted and three new observations valued $3, 4$ and $5$ are added to the data, then the mean of the resultant data is

If $12$ identical balls are to be placed in $3$ identical boxes, then the probability that one of the boxes contains exactly $3$ balls is

A factory is operating in two shifts, day and night, with $70$ and $30$ workers, respectively.If per day mean wage of the day shift workers is, $₹54$ and per day mean wage of all the workers is $₹60$, then per day mean wage of the night shift workers (in $₹$) is :

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

The variance of the first $50$ even natural numbers is :

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 :

If $X$ has a binomial distribution, $B(n, p)$ with parameters $n$ and $p$ such that $P(X=2)=P(X=3)$, then $\mathrm{E}(\mathrm{X})$, the mean of variable $\mathrm{X}$, is

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.

If $A$and $B$ are two events such that $P(A\cup B)=P(A\cap B)$, then the incorrect statement amongst the following statements is :

A number $\mathrm{x}$ is chosen at random from the set $\{1$, $2,3,4, \ldots ., 100\}$. Define the event: $\mathrm{A}=$ the chosen number $x$ satisfies $\frac{(x-10)(x-50)}{(x-30)} \geq 0$ Then $\mathrm{P}(\mathrm{A})$ is: