Probability & Statistics PYQ — Page 8

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 _____.

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

The probability that a relation $R$ from ${x,y}$ to ${x,y}$ is both symmetric and transitive, is equal to:

Let $S={E,{E}_{2}\ldots {E}_{8}}$ be a sample space of raddom experiment such that $P({E}_{n})=\frac{n}{36}$ for every $n=1,2\ldots .8$. Then the number of elements in the set ${A\subset S:P(A)\geq \frac{4}{5}}$ is _____.

The probability, that in a randomly selected $3$-digit number at least two digits are odd, is

The mean of the numbers $a,b,8,5,10$ is $6$ and their variance is $6.8$. If $M$ is the mean deviation of the numbers about the mean, then $25M$ is equal to

Let $S$ be the sample space of all five digit numbers. If $p$ is the probability that a randomly selected number from $S$, is a multiple of $7$ but not divisible by $5$, then $9p$ is equal to

A random variable $X$ has the following probability distribution: <table class="pyq-table"><tbody><tr><td>$X$</td><td>$0$</td><td>$1$</td><td>$2$</td><td>$3$</td><td>$4$</td></tr><tr><td>$P(X)$</td><td>$k$</td><td>$2k$</td><td>$4k$</td><td>$6k$</td><td>$8k$</td></tr></tbody></table>The value of $P(\frac{1<x<4}{x\leq 2})$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:

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

The mean and standard deviation of $40$ observations are $30$ and $5$ respectively. It was noticed that two of these observations $12$ and $10$ were wrongly recorded. If $\sigma$ is the standard deviation of the data after omitting the two wrong observations from the data, then $38{\sigma }^{2}$ is equal to _______.

If a point $A(x,y)$ lies in the region bounded by the $y$-axis, straight lines $2y+x=6$ and $5x-6y=30$, then the probability that $y<1$ is

Let $S={1,2,3,\ldots ,2022}$. Then the probability, that a randomly chosen number $n$ from the set $S$ such that $HCF(n,2022)=1$, is

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 _____

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

Let the mean and the variance of $20$ observations ${x}_{1},{x}_{2},\ldots {x}_{20}$ be $15$ and $9$, respectively. For $\alpha \in R$, if the mean of ${({x}_{1}+\alpha )}^{2},{({x}_{2}+\alpha )}^{2},\ldots ,{({x}_{20}+\alpha )}^{2}$ is $178$, then the square of the maximum value of $\alpha$ is equal to ______.

The mean and standard deviation of $15$ observations are found to be $8$ and $3$ respectively. On rechecking it was found that, in the observations, $20$ was misread as $5$. Then, the correct variance is equal to _____.

The mean and variance of $10$ observations were calculated as $15$ and $15$ respectively by a student who took by mistake $25$ instead of $15$ for one observation. Then, the correct standard deviation is _______.

If $A$ and $B$ are two events such that $P(A)=\frac{1}{3},P(B)=\frac{1}{5}$ and $P(A\cup B)=\frac{1}{2}$, then $P(A{B}^{'})+P(B{A}^{'})$ is equal to

The mean and standard deviation of $20$ observations were calculated as $10$ and $2.5$ respectively. It was found that by mistake one data value was taken as $25$ instead of $35.$ If $\alpha$ and $\sqrt{\beta }$ are the mean and standard deviation respectively for correct data, then $(\alpha ,\beta )$ is:

Let $A$ and $B$ be independent events such that $P(A)=p,P(B)=2p$. The largest value of $p$, for which $P$ (exactly one of $A,B$ occurs)$=\frac{5}{9}$, is:

An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first unit functions is $0.9$ and that of the second unit is $0.8.$ The instrument is switched on and it fails to operate. If the probability that only the first unit failed and second unit is functioning is $p,$ then $98p$ is equal to

The mean of first n natural numbers is:

A student appeared in an examination consisting of $8$ true-false type questions. The student guesses the answers with equal probability. The smallest value of $n,$ so that the probability of guessing at least $n$ correct answers is less than $\frac{1}{2},$ is :