JEE Probability and Statistics PYQs with Solutions PDF, Check

From the first 100 natural numbers, two numbers first a and then b are selected randomly without replacement. If the probability that $$a-b \geq 10$$ is $$\frac{m}{n}$$, gcd (m, n) = 1, then m + n is equal to______.

Two distinct numbers a and b are selected at random from 1, 2, 3, ... , 50. The probability, that their product ab is divisible by 3, is

From a lot containing 10 defective and 90 non-defective bulbs, 8 bulbs are selected one by one with replacement. Then the probability of getting at least 7 defective bulbs is

Let S be a set of 5 elements and P(S) denote the power set of S. Let E be an event of choosing an ordered pair (A, B) from the set P(S) x P(S) such that $$A\cap B=\phi.$$ If
the probability of the event E is $$\frac{3^{p}}{2^{q}}$$, where p,q $$\in$$ N, then p + q is equal to __________

If and are two events such that P(A ∩ B) = 0.1. and P(A | B) and P(B ∣ A) are the roots of the equation $$12x^{2} − 7x + 1 = 0$$, then the value of $$\frac{P(\overline{A} \cup \overline{B})}{P(\overline{A} \cap \overline{B}}$$ is:

Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $$\frac{m}{n}$$, where $$gcd(m,n)=1$$, then $$m+n$$ is equal to :

A board has 16 squares as shown in the figure:

Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is:

A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B throws a sum of 8,  and B wins if he throws a sum of 8 before A throws a sum of 5.  The probability that A wins if A makes the first throw, is:

One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5, when both the dice are thrown together, is

Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains n white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is 29/45 , then n is equal to :

Three urns A, B and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn A is:

Let S be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set S, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is :

Bag $$B_{1}$$ contains 6 white and 4 blue balls, Bag $$B_{2}$$ contains 4 white and 6 blue balls, and Bag $$B_{3}$$ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag $$B_{2}$$, is :

A bag contains $$8$$ balls, whose colours are either white or black. $$4$$ balls are drawn at random without replacement and it was found that $$2$$ balls are white and other $$2$$ balls are black. The probability that the bag contains equal number of white and black balls is:

Let Ajay will not appear in JEE exam with probability $$p = \frac{2}{7}$$, while both Ajay and Vijay will appear in the exam with probability $$q = \frac{1}{5}$$. Then the probability, that Ajay will appear in the exam and Vijay will not appear 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 _______.

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 :

A fair die is thrown until $$2$$ appears. Then the probability, that $$2$$ appears in even number of throws, is

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

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

Bag $$A$$ contains 3 white, 7 red balls and bag $$B$$ contains 3 white, 2 red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from the bag $$A$$, if the ball drawn is white, 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

Three rotten apples are accidently mixed with fifteen good apples. Assuming the random variable $$x$$ to be the number of rotten apples in a draw of two apples, the variance of $$x$$ is

A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed 3 times, then the probability of getting two tails and one head is

If the mean of the following probability distribution of a random variable $$X$$:

is $$\frac{46}{9}$$, then the variance of the distribution is

A random varaible X takes values 0,1,2,3 with probabilities $$\frac{2a+1}{30},\frac{8a-1}{30},\frac{4a+1}{30}$$, b respectively, where $$a,b \epsilon R$$. let $$\mu$$ and $$\sigma$$ respectively be the mean and standard deviation of X such that $$\sigma^{2}+\mu^{2}=2$$. Then $$\frac{a}{b}$$ is equal to :

Let the mean and variance of 7 observations 2, 4, 10, x, 12, 14, y, x >  y, be 8 and 16 respectively. Two numbers are chosen from {1, 2, 3, x - 4,y,5} one after an other without replacement, then the probability, that the smaller number among the two chosen numbers is less than 4, is :

lf the mean deviation about the median of the numbers k, 2k, 3k, ..... , 1000k is 500, then $$k^{2}$$ is equal to :

Let the mean and variance of 8 numbers - 10, - 7, - 1, x, y, 9, 2, 16 be $$\frac{7}{2}\text{ and }\frac{293}{4}$$ respectively.
Then the mean of 4 numbers x, y, x + y + 1, |x-y| is:

If a random variable x has the probability distribution

then $$ P(3< x\leq 6)$$ is equal to

The mean and variance of a data of 10 observations are 1O and 2, respectively. If an observations $$\alpha$$ in this data is replaced by $$\beta$$, then the mean and variance become 10.1 and 1.99, respectively. Then $$\alpha+\beta$$ equals

Let $$X= \left\{x\in N:1\leq x\leq19 \right\}$$ and for some $$a,b \in \mathbb R, Y = \left\{ax+b:x\in X\right\}.$$ If the mean and variance of the elements of Y are 30 and 750, respectively, then the sum of all possible values of b is

The mean and variance of 10 observations are 9 and 34.2, respectively. If 8 of these observations are 2, 3, 5, 10, 11 , 13, 15, 21, then the mean deviation about the median of all the 10 observations is

If the mean and the variance of the data

are $$ \mu $$ and 19 respectively, then the value of $$\lambda$$ $$+\mu$$ is

The probability distribution of a random variable X is given below:

If $$ E(X)=\frac{263}{15} $$. then $$ P(X<20)$$ is equal to:

A coin is tossed three times. Let  $$X$$ denote the number of times a tail follows a head. If $$\mu$$ and $$\sigma^{2}$$ denote the mean and variance of $$X$$, then the value of $$64(\mu+\sigma^{2})$$ is :

The variance of the numbers 8, 21, 34, 47,…, 320 is

For a statistical data $$x_1,x_2,\ldots,x_{10}$$ of 10 values, a student obtained the mean as  5.5 and  $$\sum_{i=1}^{10} x_i^2 = 371. $$  He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values  6 and 8  respectively. The variance of the corrected data is:

Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If $$x$$ denote the number of defective oranges, then the variance of $$x$$ is

Let $$x_{1},x_{2},...x_{10}$$ be ten observations such that $$\sum_{i=1}^{10}(x_{i}-2)=30,\sum_{i=1}^{10}(x_{i}-\beta)^{2}=98,\beta > 2$$, and their variance is $$\frac{4}{5}$$. If $$\mu$$ and $$\sigma^{2}$$ are respectively the mean and the variance of $$2(x_{1}-1)+4\beta, 2(x_{2}-1)+4\beta,....,2(x_{10}-1)+4\beta$$, then $$\frac{\beta \mu}{\sigma^{2}}$$ is equal to :

Marks obtains by all the students of class 12 are presented in a freqency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12 . If the number of students whose marks are less than 12 is 18 , then the total number of students is

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

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

Consider 10 observations $$x_1, x_2, \ldots, x_{10}$$, 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. Then $$\frac{\beta}{\alpha}$$ is equal to:

Let $$a_1, a_2, \ldots, a_{10}$$ be 10 observations such that $$\sum_{k=1}^{10} a_k = 50$$ and $$\sum_{\forall k < j} a_k \cdot a_j = 1100$$. Then the standard deviation of $$a_1, a_2, \ldots, a_{10}$$ is equal to :

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 _____.

If the mean and variance of the data $$65, 68, 58, 44, 48, 45, 60, \alpha, \beta, 60$$ where $$\alpha > \beta$$ are $$56$$ and $$66.2$$ respectively, then $$\alpha^2 + \beta^2$$ is equal to _______

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 is equal to

Let $$M$$ denote the median of the following frequency distribution.

Then $$20M$$ is equal to :

If the variance $$\sigma^2$$ of the data

is $$k$$, then the value of $$k$$ is ______ (where $$.$$ denotes the greatest integer function)