Probability & Statistics PYQ — Page 8

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 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

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

The mean and variance of the data $4,5,6,6,7,8,x,y$ where $x<y$ are $6$ and $\frac{9}{4}$ respectively. Then ${x}^{4}+{y}^{2}$ is equal to

Suppose a class has $7$ students. The average marks of these students in the mathematics examination is $62$, and their variance is $20$. A student fails in the examination if he/she gets less than $50$ marks, then in worst case, the number of students can fail is

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

Let ${E}_{1}$ and ${E}_{2}$ be two events such that the conditional probabilities $P({E}_{1}\mid {E}_{2})=\frac{1}{2}$, $P({E}_{2}\mid {E}_{1})=\frac{3}{4}$ and $P({E}_{1}\cap {E}_{2})=\frac{1}{8}$. Then

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

Five numbers ${x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5}$ are randomly selected from the numbers$1,2,3,\ldots \ldots ,18$ and are arranged in the increasing order $({x}_{1}<{x}_{2}<{x}_{1}<{x}_{4}<{x}_{2})$. The probability that ${x}_{2}=7$ and ${x}_{4}=11$ is

If P(A)=0.4, P(B)=0.5 and P(A∩B)=0.2 then P(A|B) is:

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 _______.

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

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 _____.

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

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

A six faced die is biased such that $3\times P$(a prime number)$=6\times P$(a composite number)$=2\times P(1)$. Let $X$ be a random variable that counts the number of times one gets a perfect square on some throws of this die. If the die is thrown twice, then the mean of $X$ is

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:

A fair coin is tossed $n-$ times such that the probability of getting at least one head is at least $0.9.$ Then the minimum value of $n$ 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

Let the mean and variance of the frequency distribution <table class="pyq-table"><tbody><tr><td>$x:$</td><td>${x}_{1}=2$</td><td>${x}_{2}=6$</td><td>${x}_{3}=8$</td><td>${x}_{4}=9$</td></tr><tr><td>$f:$</td><td>$4$</td><td>$4$</td><td>$\alpha$</td><td>$\beta$</td></tr></tbody></table>be $6$ and $6.8$ respectively. If ${x}_{3}$ is changed from $8$ to $7,$ then the mean for the new data will be:

A fair die is tossed until six is obtained on it. Let $X$ be the number of required tosses, then the conditional probability $P(X\geqslant 5\mid X>2)$ is :