Probability & Statistics PYQ — Page 7

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

Two dice are thrown independently. Let $A$ be the event that the number appeared on the ${1}^{\text{st }}$ die is less than the number appeared on the ${2}^{\text{nd }}$ die, $B$ be the event that the number appeared on the ${1}^{\text{st }}$ die is even and that on the second die is odd, and $C$ be the event that the number appeared on the ${1}^{\text{st }}$ die is odd and that on the ${2}^{\text{nd }}$ is even. Then

Let the mean of 6 observations $1,2,4,5,x$ and $y$ be $5$ and their variance be $10$. Then their mean deviation about the mean is equal to

Let $N$ denote the sum of the numbers obtained when two dice are rolled. If the probability that ${2}^{N}<N!$ is $\frac{m}{n}$ where $m$ and $n$ are coprime, then $4m-3n$ 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 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 ____________.

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

The mean and standard deviation of the marks of $10$ students were found to be $50$ and $12$ respectively. Later, it was observed that two marks $20$ and $25$ were wrongly read as $45$ and $50$ respectively. Then the correct variance is

Let the mean and variance of $8$ numbers $x,y,10,12,6,12,4,8$ be $9$ and $9.25$ respectively. If $x>y,$ then $3x-2y$ is equal to $_______$

Let $A$ and $B$ be two events such that $P(B\mid A)=\frac{2}{5}$, $P(A\mid B)=\frac{1}{7}$ and $P(A\cap B)=\frac{1}{9}$. Consider $(S1)P({A}^{'}\cup B)=\frac{5}{6}$, $(S2)P({A}^{'}\cap {B}^{'})=\frac{1}{18}$. Then

If the numbers appeared on the two throws of a fair six faced die are $\alpha$ and $\beta$, then the probability that ${x}^{2}+\alpha x+\beta >0$, for all $x\in R$, is

Out of $60%$ female and $40%$ male candidates appearing in an exam, $60%$ candidates qualify it. The number of females qualifying the exam is twice the number of males qualifying it. A candidate is randomly chosen from the qualified candidates. The probability, that the chosen candidate is a female, is

If the mean deviation about the mean of the numbers $1,2,3,\ldots \ldots ,n$, where $n$ is odd, is $\frac{5(n+1)}{n}$, then $n$ is equal to ______.

Let the mean and the variance of $5$ observations ${x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5}$ be $\frac{24}{5}$ and $\frac{194}{25}$ respectively. If the mean and variance of the first $4$ observation are $\frac{7}{2}$ and $a$ respectively, then $(4a+{x}_{5})$ is equal to

Let ${E}_{1},{E}_{2},{E}_{3}$ be three mutually exclusive events such that $P({E}_{1})=\frac{2+3p}{6},P({E}_{2})=\frac{2-p}{8}$ and $P({E}_{3})=\frac{1-p}{2}$. If the maximum and minimum values of $p$ are ${p}_{1}$ and ${p}_{2}$ then $({p}_{1}+{p}_{2})$ is equal to:

If the mean deviation about median for the number $3,5,7,2k,12,16,21,24$ arranged in the ascending order, is $6$ then the median is

If the probability that a randomly chosen $6$-digit number formed by using digits $1$ and $8$ only is a multiple of $21$ is $p$, then $96p$ is equal to _____.

Let the mean of $50$ observations is $15$ and the standard deviation is $2$. However, one observation was wrongly recorded. The sum of the correct and incorrect observations is $70$. If the mean of the correct set of observations is $16$, then the variance of the correct set is equal to

Bag $A$ contains $2$ white, $1$ black and $3$ red balls and bag $B$ contains $3$ black, $2$ red and $n$ white balls. One bag is chosen at random and $2$ balls drawn from it at random are found to be $1$ red and $1$ black. If the probability that both balls come from Bag $A$ is $\frac{6}{11}$, then $n$ is equal to _____

A bag contains $4$ white and $6$ black balls. Three balls are drawn at random from the bag. Let $X$ be the number of white balls, among the drawn balls. If ${\sigma }^{2}$ is the variance of $X$, then $100{\sigma }^{2}$ is equal to

The number of values of $a\in N$ such that the variance of $3,7,12,a,43-a$ is a natural number is:

Bag $I$ contains $3$ red, $4$ black and $3$ white balls and Bag $\mathrm{II}$ contains $2$ red, $5$ black and $2$ white balls. One ball is transferred from Bag $I$ to Bag $\mathrm{II}$ and then a ball is draw from Bag $\mathrm{II}$. The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red, is

A biased die is marked with numbers $2,4,8,16,32,32$ on its faces and the probability of getting a face with mark $n$ is $\frac{1}{n}$. If the die is thrown thrice, then the probability, that the sum of the numbers obtained is $48$, is

In an examination, there are $10$ true-false type questions. Out of $10$, a student can guess the answer of $4$ questions correctly with probability $\frac{3}{4}$ and the remaining $6$ questions correctly with probability $\frac{1}{4}$. If the probability that the student guesses the answers of exactly $8$ questions correctly out of $10$ is $\frac{27k}{{4}^{10}}$, then $k$ is equal to