Probability & Statistics PYQ — Page 4

In a binomial distribution with n=5 and p=1/3, the probability of exactly 2 successes is:

An urn contains $6$ white and $9$ black balls. Two successive draws of $4$ balls are made without replacement. The probability, that the first draw gives all white balls and the second draw gives all black balls, is :

The coefficients $a, b, c$ in the quadratic equation $a x^2+b x+c=0$ are from the set $\{1,2,3,4,5,6\}$. If the probability of this equation having one real root bigger than the other is $p$, then 216 p equals :

Let $\mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathrm{N}$ and $\mathrm{a} < \mathrm{b} < \mathrm{c}$. Let the mean, the mean deviation about the mean and the variance of the 5 observations $9,25, \mathrm{a}, \mathrm{b}, \mathrm{c}$ be 18,4 and $\frac{136}{5}$, respectively. Then $2 \mathrm{a}+\mathrm{b}-\mathrm{c}$ is equal to__________

An integer is chosen at random from the integers $1,2,3,.....,50$. The probability that the chosen integer is a multiple of atleast one of $4,6$ and $7$ is

Let the median and the mean deviation about the median of $7$ observation $170,125,230,190,210,a,b$ be $170$ and $\frac{205}{7}$ respectively. Then the mean deviation about the mean of these $7$ observations is:

The mean and standard deviation of $15$ observations were found to be $12$ and $3$ respectively. On rechecking it was found that an observation was read as $10$ in place of $12.$ If $\mu$ and ${\sigma }^{2}$ denote the mean and variance of the correct observations respectively, then $15(\mu +{\mu }^{2}+{\sigma }^{2})$ is equal to _________.

Let the mean and the variance of $6$ observation $a,b,68,44,48,60$ be $55$ and $194$, respectively if $a>b$, then $a+3b$ is

If the mean and variance of five observations are $\frac{24}{5}$ and $\frac{194}{25}$ respectively and the mean of first four observations is $\frac{7}{2}$, then the variance of the first four observations in equal to

If the variance ${\sigma }^{2}$ of the data $\begin{matrix}{x}_{i} & 0 & 1 & 5 & 6 & 10 & 12 & 17 \\ {f}_{i} & 3 & 2 & 3 & 2 & 6 & 3 & 3\end{matrix}$ is $k$ then the value of $[k]$ is ______ {where $[.]$ denotes the greatest integer funciton}

A bag contains 5 red and 3 blue balls. Two balls are drawn at random without replacement. The probability that both are red is:

Let $\alpha, \beta \in \mathbf{R}$. Let the mean and the variance of 6 observations $-3,4,7,-6, \alpha, \beta$ be 2 and 23 , respectively. The mean deviation about the mean of these 6 observations is :

Two marbles are drawn in succession from a box containing $10$ red, $30$ white, $20$ blue and $15$ orange marbles, with replacement being made after each drawing. Then the probability, that first drawn marble is red and second drawn marble is white, is

Let the mean and the standard deviation of the probability distribution $\begin{array}{|c|c|c|c|c|} \hline \mathrm{X} & \mathrm{\alpha} & \mathrm{1} & 0 & -3 \\ \hline \mathrm{P}(\mathrm{X}) & \frac{1}{3} & \mathrm{~K} & \frac{1}{6} & \frac{1}{4} \\ \hline \end{array}$ be $\mu$ and $\sigma$, respectively. If $\sigma-\mu=2$, then $\sigma+\mu$ is equal to________

From a lot of 10 items, which include 3 defective items, a sample of 5 items is drawn at random. Let the random variable $\mathrm{X}$ denote the number of defective items in the sample. If the variance of $\mathrm{X}$ is $\sigma^2$, then $96 \sigma^2$ is equal to ______

Let the sum of two positive integers be 24 . If the probability, that their product is not less than $\frac{3}{4}$ times their greatest possible product, is $\frac{m}{n}$, where $\operatorname{gcd}(m, n)=1$, then $n-m$ equals

Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables $X$ and $Y$ respectively denote the number of blue and yellow balls. If $\bar{X}$ and $\bar{Y}$ are the means of $X$ and $Y$ respectively, then $7 \bar{X}+4 \bar{Y}$ is equal to________

The mean and standard deviation of 20 observations are found to be 10 and 2 . respectively. On rechecking, it was found that an observation by mistake was taken 8 instead of 12. The correct standard deviation is

Let ${a}_{1},{a}_{2},...,{a}_{10}$ be $10$ observations such that $\sum _{k=1}^{10}{a}_{k}=50$ and $\underset{\forall k<j}{\sum }{a}_{k}\cdot {a}_{j}=1100$. Then the standard deviation of ${a}_{1},{a}_{2},\ldots ,{a}_{10}$ is equal to :

Consider $10$ observation ${x}_{1},{x}_{2},...{x}_{10}$, such that $\sum _{i=1}^{10}({x}_{i}-\alpha )=2$ and $\sum _{i=1}^{10}{({x}_{i}-\beta )}^{2}=40$, where $\alpha ,\beta$ are positive integers. Let the mean and the variance of the observations be $\frac{6}{5}$ and $\frac{84}{25}$ respectively. The $\frac{\beta }{\alpha }$ is equal to:

Two integers $x\text{and}y$ are chosen with replacement from the set ${0,1,2,3,\ldots ..,10}$. Then the probability that $|x-y|>5$ is :

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as $\frac{1}{3}$ and $\frac{2}{3}$ respectively. Let $x$ be the number of matches that the team wins, and $y$ be the number of matches that team loses. If the probability $\mathrm{P}(|x-y| \leq$ 2) is $p$, then $3^9 p$ equals ______

If the mean of the following probability distribution of a random variable $\mathrm{X}$ : $\begin{array}{|c|c|c|c|c|c|} \hline \mathrm{X} & 0 & 2 & 4 & 6 & 8 \\ \hline \mathrm{P}(\mathrm{X}) & a & 2 a & a+b & 2 b & 3 b \\ \hline \end{array}$ is $\frac{46}{9}$, then the variance of the distribution is

A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required and let $a=P(X=3),b=P(X\geq 3)$ and $c=$ $P(X\geq 6\mid X>3)$. Then $\frac{b+c}{a}$ is equal to