Probability & Statistics PYQ — Page 10

Let $X$ be a random variable such that the probability function of a distribution is given by $P(X=0)=\frac{1}{2},P(X=j)=\frac{1}{{3}^{j}}(j=1,2,3,\ldots ,\infty ).$ Then the mean of the distribution and $P(X$ is positive and even$)$ respectively, are:

Consider three observations $a,b$ and $c$ such that $b=a+c$. If the standard deviation of $a+2,c+2$ is $d$, then which of the following is true?

Let $A,B$ and $C$ be three events such that the probability that exactly one of $A$ and $B$ occurs is $(1-k),$ the probability that exactly one of $B$ and $C$ occurs is $(1-2k),$ the probability that exactly one of $C$ and $A$ occurs is $(1-k)$ and the probability of all $A,B$ and $C$ occur simultaneously is ${k}^{2},$ where $0<k<1.$ Then the probability that at least one of $A,B$ and $C$ occur is:

The probability that a randomly selected $2-$digit number belongs to the set ${n\in N:({2}^{n}-2)$ is a multiple of $3}$ is equal to

Let $n$ be an odd natural number such that the variance of $1,2,3,4,\ldots ,n$ is $14.$ Then $n$ is equal to ________.

The coefficients $a,b$ and $c$ of the quadratic equation, $a{x}^{2}+bx+c=0$ are obtained by throwing a dice three times. The probability that this equation has equal roots is:

Consider a set of $3n$ numbers having variance $4.$ In this set, the mean of first $2n$ numbers is $6$ and the mean of the remaining $n$ numbers is $3.$ A new set is constructed by adding $1$ into each of the first $2n$ numbers, and subtracting $1$ from each of the remaining $n$ numbers. If the variance of the new set is $k,$ then $9k$ is equal to ______.

When a certain biased die is rolled, a particular face occurs with probability $\frac{1}{6}-x$ and its opposite face occurs with probability $\frac{1}{6}+x.$ All other faces occur with probability $\frac{1}{6}.$ Note that opposite faces sum to $7$ in any die. If $0<x<\frac{1}{6},$ and the probability of obtaining total sum $=7,$ when such a die is rolled twice, is $\frac{13}{96},$ then the value of $x$ is

Let the mean and variance of four numbers $3,7,x$ and $y(x>y)$ be $5$ and $10$ respectively. Then the mean of four numbers $3+2x,7+2y,x+y$ and $x-y$ is ______.

Consider the following frequency distribution : <table class="pyq-table"><tbody><tr><td>class</td><td>$10-20$</td><td>$20-30$</td><td>$30-40$</td><td>$40-50$</td><td>$50-60$</td></tr><tr><td>Frequency</td><td>$\alpha$</td><td>$110$</td><td>$54$</td><td>$30$</td><td>$\beta$</td></tr></tbody></table>If the sum of all frequencies is $584$ and median is $45$, then $|\alpha -\beta |$ is equal to .

Let $X$ be a random variable with distribution. <table class="pyq-table"><tbody><tr><td>$x$</td><td>$-2$</td><td>$-1$</td><td>$3$</td><td>$4$</td><td>$6$</td></tr><tr><td>$P(X=x)$</td><td>$\frac{1}{5}$</td><td>$a$</td><td>$\frac{1}{3}$</td><td>$\frac{1}{5}$</td><td>$b$</td></tr></tbody></table>If the mean of $X$ is $2.3$ and variance of $X$ is ${\sigma }^{2},$ then $100{\sigma }^{2}$ is equal to :

Let $9$ distinct balls be distributed among $4$ boxes, ${B}_{1},{B}_{2},{B}_{3}$ and ${B}_{4}$. If the probability that ${B}_{3}$ contains exactly $3$ balls is $k{(\frac{3}{4})}^{9}$ then $k$ lies in the set :

If the variance of $10$ natural numbers $1,1,1,\ldots ,1,k$ is less than $10,$ then the maximum possible value of $k$ is ___________.

A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is :

Consider the following frequency distribution: <table class="pyq-table"><tbody><tr><td>Class:</td><td>$0-6$</td><td>$6-12$</td><td>$12-18$</td><td>$18-24$</td><td>$24-30$</td></tr><tr><td>Frequency:</td><td>$a$</td><td>$b$</td><td>$12$</td><td>$9$</td><td>$5$</td></tr></tbody></table>If mean $=\frac{309}{22}$ and median $=14,$ then the value $(a-b{)}^{2}$ is equal to

Let ${X}_{1},{X}_{2},\ldots ,{X}_{18}$ be eighteen observations such that $\sum _{i=1}^{18}({X}_{i}-\alpha )=36$ and $\sum _{i=1}^{18}{({X}_{i}-\beta )}^{2}=90$, where $\alpha$ and $\beta$ are distinct real numbers. If the standard deviation of these observations is $1$, then the value of $|\alpha -\beta |$ is _______.

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:

Consider the statistics of two sets of observations as follows: <table class="pyq-table"><tbody><tr><td></td><td>Size</td><td>Mean</td><td>Variance</td></tr><tr><td>Observation I</td><td>$10$</td><td>$2$</td><td>$2$</td></tr><tr><td>Observation II</td><td>$n$</td><td>$3$</td><td>$1$</td></tr></tbody></table>If the variance of the combined set of these two observations is $\frac{17}{9},$ then the value of $n$ is equal to ________.

Let $S={1,2,3,4,5,6}.$ Then the probability that a randomly chosen onto function $g$ from $S$ to $S$ satisfies $g(3)=2g(1)$ is :

Let a computer program generate only the digits $0$ and $1$ to form a string of binary numbers with probability of occurrence of $0$ at even places be $\frac{1}{2}$ and probability of occurrence of $0$ at the odd place be $\frac{1}{3}.$ Then the probability that $10$ is followed by $01$ is equal to :

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 _______.

The probabilities of three events $A,B$ and $C$ are given $P(A)=0.6,P(B)=0.4$ and $P(C)=0.5$. If $P(A\cup B)=0.8,P(A\cap C)=0.3,P(A\cap B\cap C)=0.2,P(B\cap C)=\beta$ and $P(A\cup B\cup C)=\alpha$, where $0.85\leq \alpha \leq 0.95,$ then $\beta$ lies in the interval :

In a game two players $A$ and $B$ take turns in throwing a pair of fair dice starting with player $A$ and total of scores on the two dice, in each throw is noted. $A$ wins the game if he throws a total of $6$ before $B$ throws a total of $7$ and $B$ wins the game if he throws a total of $7$ before $A$ throws a total of six. The game stops as soon as either of the players wins. The probability of $A$ winning the game is :

Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up a three or a five, is