Probability & Statistics PYQ — Page 16

Four numbers are chosen at random (without replacement) from the set $\{1,2,3, \ldots ., 20\}$. Statement-1: The probability that the chosen numbers when arranged in some order will form an AP is $\frac{1}{85}$. Statement-2: If the four chosen numbers from an AP, then the set of all possible values of common difference is $\{\pm 1, \pm 2, \pm 3, \pm 4, \pm 5\}$.

For two data sets, each of size 5 , the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4 , respectively. The variance of the combined data set is

If the mean deviation of number $1,1+d, 1+2 d, \ldots ., 1+100 d$ from their mean is 255 , then the $d$ is equal to

Statement-1 : The variance of first $n$ even natural numbers is $\frac{n^2-1}{4}$ Statement-2 : The sum of first $n$ natural numbers is $\frac{n(n+1)}{2}$ and the sum of squares of first $n$ natural numbers is $\frac{n(n+1)(2 n+1)}{6}$

One ticket is selected at random from 50 tickets numbered $00,01,02, \ldots, 49$. Then the probability that the sum of the digits on the selected ticket is 8 , given that the product of these digits is zero, equals

A die is thrown. Let $A$ be the event that the number obtained is greater than 3 . Let $B$ be the event that the number obtained is less than 5 . Then $P(A \cup B)$ is

It is given that the events $A$ and $B$ are such that $P(A)=\frac{1}{4}, P\left(\frac{A}{B}\right)=\frac{1}{2}$ and $P\left(\frac{B}{A}\right)=\frac{2}{3}$. Then $P(B)$ is

The mean of the numbers $a, b, 8,5,10$ is 6 and the variance is $6.80$. Then which one of the following gives possible values of $a$ and $b$ ?

A pair of fair dice is thrown independently three times. The probability of getting a score of exactly $9$ twice is

The average marks of boys in a class is $52$ and that of girls is $42$. The average marks of boys and girls combined is $50$. The percentage of boys in the class is

At a telephone enquiry system the number of phone cells regarding relevant enquiry follow Poisson distribution with an average of 5 phone calls during 10-minute time intervals. The probability that there is at the most one phone call during a 10-minute time period is

Suppose a population $A$ has 100 observations $101,102, \ldots, 200$, and another population $B$ has 100 observations $151,152, \ldots, 250$. If $V_A$ and $V_B$ represent the variances of the two populations, respectively, then $\frac{V_A}{V_B}$ is

Let $\mathrm{x}_1, \mathrm{x}_2, \ldots, \mathrm{x}_{\mathrm{n}}$ be $\mathrm{n}$ observations such that $\sum \mathrm{x}_{\mathrm{i}}^2=400$ and $\sum \mathrm{x}_{\mathrm{i}}=80$. Then a possible value of $\mathrm{n}$ among the following is

A random variable $X$ has Poisson distribution with mean 2. Then $P(X>1.5)$ equals

Let $A$ and $B$ be two events such that $P(\overline{A \cup B})=\frac{1}{6}, P(A \cap B)=\frac{1}{4}$ and $P(\bar{A})=\frac{1}{4}$, where $\bar{A}$ stands for complement of event $A$. Then events $A$ and $B$ are

Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is

If in a frequently distribution, the mean and median are 21 and 22 respectively, then its mode is approximately

A random variable $X$ has the probability distribution: \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \mathrm{X}: & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \mathrm{p}(\mathrm{X}): & 0.15 & 0.23 & 0.12 & 0.10 & 0.20 & 0.08 & 0.07 & 0.05 \\ \hline \end{array} For the events $E=\{X$ is a prime number $\}$ and $F=\{X < 4\}$, the probability $P(E \cup F)$ is

Consider the following statements: Mode can be computed from histogram Median is not independent of change of scale Variance is independent of change of origin and scale.

In a series of $2 n$ observations, half of them equal a and remaining half equal $-a$. If the standard deviation of the observations is 2 , then $|a|$ equals

The probability that A speaks truth is $\frac{4}{5}$, while this probability for $B$ is $\frac{3}{4}$. The probability that they contradict each other when asked to speak on a fact is

The median of a set of 9 distinct observations is $20.5$. If each of the largest 4 observations of the set is increased by 2 , then median of the new set

In an experiment with 15 observations on $x$, the following results were available: $\Sigma x^2=2830, \Sigma x=170$ One observation that was 20 was found to be wrong and was replaced by the correct value 30 . The corrected variance is

Events $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are mutually exclusive events such that $\mathrm{P}(\mathrm{A})=\frac{3 \mathrm{x}+1}{3}, \mathrm{P}(\mathrm{B})=\frac{\mathrm{x}-1}{4}$ and $\mathrm{P}(\mathrm{C})=\frac{1-2 \mathrm{x}}{4}$. The set of possible values of $\mathrm{x}$ are in the interval.