Probability — JEE Main PYQs

A man throws a fair coin repeatedly. He gets 10 points for each head he throws and 5 points for each tail he throws. If the probability that he gets exactly 30 points is $\frac{m}{n}$, $\gcd(m, n) = 1$, then $m + n$ is equal to :

A coin is tossed 8 times. If the probability that exactly 4 heads appear in the first six tosses and exactly 3 heads appear in the last five tosses is $p$, then $96p$ is equal to ______.

From a month of 31 days, 3 different dates are selected at random. If the probability that these dates are in an increasing A.P. is equal to $\dfrac{a}{b}$, where $a, b \in \mathbb{N}$ and $\gcd(a, b) = 1$, then $a + b$ is equal to __________

A letter is known to have arrived by post either from KANPUR or from ANANTPUR. On the envelope just two consecutive letters AN are visible. The probability, that the letter came from ANANTPUR, is:

The probabilities that players A and B of a team are selected for the captaincy for a tournament are 0.6 and 0.4, respectively. If A is selected the captain, the probability that the team wins the tournament is 0.8 and if B is selected the captain, the probability that the team wins the tournament is 0.7. Then the probability, that the team wins the tournament, is :

A bag contains $(N+1)$ coins $-$ $N$ fair coins, and one coin with 'Head' on both sides. A coin is selected at random and tossed. If the probability of getting 'Head' is $\frac{9}{16}$, then $N$ is equal to:

A bag contains 6 blue and 6 green balls. Pairs of balls are drawn without replacement until the bag is empty. The probability that each drawn pair consists of one blue and one green ball is :

A candidate has to go to the examination centre to appear in an examination. The candidate uses only one means of transportation for the entire distance out of bus, scooter and car. The probabilities of the candidate going by bus, scooter and car, respectively, are $\frac{2}{5}$, $\frac{1}{5}$ and $\frac{2}{5}$. The probabilities that the candidate reaches late at the examination centre are $\frac{1}{5}$, $\frac{1}{3}$ and $\frac{1}{4}$ if the candidate uses bus, scooter and car, respectively. Given that the candidate reached late at the examination centre, the probability that the candidate travelled by bus is :

Three distinct numbers are selected randomly from the set $\{1, 2, 3, \ldots, 40\}$. If the probability, that the selected numbers are in an increasing G.P., is $\frac{\mathrm{m}}{\mathrm{n}}$, $\gcd(\mathrm{m}, \mathrm{n}) = 1$, then $\mathrm{m} + \mathrm{n}$ is equal to __________.

Given three indentical bags each containing 10 balls, whose colours are as follows :

	Red	Blue	Green
Bag I	3	2	5
Bag II	4	3	3
Bag III	5	1	4

A person chooses a bag at random and takes out a ball. If the ball is Red, the probability that it is from bag I is p and if the ball is Green, the probability that it is from bag III is q, then the value of $\left(\frac{1}{p}+\frac{1}{q}\right)$ is :

All five letter words are made using all the letters A, B, C, D, E and arranged as in an English dictionary with serial numbers. Let the word at serial number $n$ be denoted by $\mathrm{W_n}$. Let the probability $\mathrm{P(W_n)}$ of choosing the word $\mathrm{W_n}$ satisfy $\mathrm{P(W_n)} = 2\mathrm{P(W_{n-1})}$, $n > 1$.

If $\mathrm{P(CDBEA)} = \frac{2^{\alpha}}{2^{\beta} - 1}$, $\alpha, \beta \in \mathbb{N}$, then $\alpha + \beta$ is equal to : __________

If the probability that the random variable $X$ takes the value $x$ is given by $P(X = x) = k(x + 1)3^{-x},\ x = 0, 1, 2, 3\ldots$, where $k$ is a constant, then $P(X \geq 3)$ is equal to

The probability, of forming a 12 persons committee from 4 engineers, 2 doctors and 10 professors containing at least 3 engineers and at least 1 doctor, is

A box contains 10 pens of which 3 are defective. A sample of 2 pens is drawn at random and let $X$ denote the number of defective pens. Then the variance of $X$ is

A card from a pack of 52 cards is lost. From the remaining 51 cards, $\mathrm{n}$ cards are drawn and are found to be spades. If the probability of the lost card to be a spade is $\dfrac{11}{50}$, then $\mathrm{n}$ is equal to __________.

Let a random variable $X$ take values $0, 1, 2, 3$ with $P(X = 0) = P(X = 1) = p$, $P(X = 2) = P(X = 3)$ and $E(X^2) = 2E(X)$. Then the value of $8p - 1$ is :

A bag contains 19 unbiased coins and one coin with head on both sides. One coin drawn at random is tossed and head turns up. If the probability that the drawn coin was unbiased, is $\dfrac{m}{n}$, $\gcd(m, n) = 1$, then $n^2 - m^2$ is equal to :

If $A$ and $B$ are two events such that $P(A) = 0.7$, $P(B) = 0.4$ and $P\left(A \cap \overline{B}\right) = 0.5$, where $\overline{B}$ denotes the complement of $B$, then $P\left(B \mid \left(A \cup \overline{B}\right)\right)$ 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 :

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 :

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

A board has 16 squares as shown in the figure :

[Figure: a 4 × 4 grid of 16 squares]

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

Let $\mathrm{A} = [a_{ij}]$ be a square matrix of order 2 with entries either 0 or 1. Let E be the event that A is an invertible matrix. Then the probability P(E) 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

Two numbers $k_1$ and $k_2$ are randomly chosen from the set of natural numbers. Then the probability that the value of $i^{k_1} + i^{k_2}$, $(i = \sqrt{-1})$ is non-zero, equals

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 select… [remainder of stem corrupted in the source PDF; the Hindi duplicate is also lost]

Bag $\mathrm{B}_1$ contains 6 white and 4 blue balls, Bag $\mathrm{B}_2$ contains 4 white and 6 blue balls, and Bag $\mathrm{B}_3$ 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 $\mathrm{B}_2$, is :

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 :

If the mean of the following probability distribution of a radam variable X:

X	0	2	4	6	8
P(X)	$a$	$2a$	$a+b$	$2b$	$3b$

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

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as $\dfrac{1}{3}$ and $\dfrac{2}{3}$ 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 $\mathrm{P}(|x - y| \le 2)$ is $p$, then $3^9 p$ equals ___________.

The coefficients a, b, c in the quadratic equation $ax^2 + bx + c = 0$ are chosen from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$. The probability of this equation having repeated roots is :

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

The coefficients $a$, $b$, $c$ in the quadratic equation $ax^2 + bx + c = 0$ 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 $216p$ equals :

Let the mean and the standard deviation of the probability distribution

$X$	$\alpha$	$1$	$0$	$-3$
$P(X)$	$\dfrac{1}{3}$	$K$	$\dfrac{1}{6}$	$\dfrac{1}{4}$

be $\mu$ and $\sigma$, respectively. If $\sigma - \mu = 2$, then $\sigma + \mu$ is equal to ________.

A company has two plants $A$ and $B$ to manufacture motorcycles. 60% motorcycles are manufactured at plant $A$ and the remaining are manufactured at plant $B$. 80% of the motorcycles manufactured at plant $A$ are rated of the standard quality, while 90% of the motorcycles manufactured at plant $B$ are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If $p$ is the probability that it was manufactured at plant $B$, then $126p$ is

If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is :

From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random variable $X$ denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If variance of $X$ is $\dfrac{\mathrm{m}}{\mathrm{n}}$, where $\gcd(\mathrm{m, n})=1$, then $\mathrm{n-m}$ 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 $\gcd(m,n)=1$, 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 __________.

There are three bags X, Y and Z. Bag X contains 5 one-rupee coins and 4 five-rupee coins; Bag Y contains 4 one-rupee coins and 5 five-rupee coins and Bag Z contains 3 one-rupee coins and 6 five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability, that it came from bag Y, is :

Let a, b and c denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked 1, 2, 3, 4. If the probability that $ax^2+bx+c=0$ has all real roots is $\frac{\mathrm{m}}{\mathrm{n}}$, $\gcd(\mathrm{m}, \mathrm{n})=1$, then $\mathrm{m}+\mathrm{n}$ is equal to ________.

If an unbiased dice is rolled thrice, then the probability of getting a greater number in the $i^{\text{th}}$ roll than the number obtained in the $(i-1)^{\text{th}}$ roll, $i=2, 3$, is equal to

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 $\dfrac{b + c}{a}$ is equal to ________.

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

Two integers $x$ and $y$ are chosen with replacement from the set $\{0, 1, 2, 3, ..., 10\}$. 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

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

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

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