Probability — JEE Main PYQs

Let $\mathrm{a},\mathrm{b}$ 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 $a{x}^2+bx+c=0$ has all real roots is $\frac{m}{n}$, …

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 $(\mathrm{i}-1{)}^{\text{th }}$ roll, $i=2,3$, is equal to :

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

A coin is based 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-

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

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

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

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

Three dice are rolled. If the probability of getting different numbers on the three dice is $\frac{p}{q}$, where $p$ and q are co-prime, then $q-p$ is equal to

In a bolt factory, machines A, B and C manufacture respectively $20\%$, $30\%$ and $50\%$ of the total bolts. Of their output 3, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found the defective, then the probab …

A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If the probability of at least 4 successes is $\frac{\mathrm{k}}{3^{11}}$,then k is equal to

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

Let a die be rolled n times. Let the probability of getting odd numbers seven times be equal to the probability of getting odd numbers nine times. If the probability of getting even numbers twice is $\frac{\mathrm{k}}{2^{15}}$, then k is equal to :

Let N denotes the sum of the numbers obtained when two dice are rolled. If the probability that $2^{\mathrm{N}}<\mathrm{N}!$ is $\frac{m}{n}$, where m and n are coprime, then $4m-3n$ is equal to

Two dice A and B are rolled. Let the numbers obtained on A and B be $\alpha$ and $\beta$ respectively. If the variance of $\alpha -\beta$ is $\frac{p}{q}$, where $p$ and $q$ are coprime, then the sum of the positive divisors of p is equal to

A fair $\mathrm{n}(\mathrm{n}>1)$ faces die is rolled repeatedly until a number less than n appears. If the mean of the number of tosses required is $\frac{\mathrm{n}}{9}$, then n is equal to

The random variable $X$ follows binomial distribution $B(n,p)$ for which the difference of the mean and the variance is 1. If $2P(X=2)=3P(X=1)$, then $n^{2}P(X>1)$ is equal to

A coin is biased so that the head is 3 times as likely to occur as tail. This coin is tossed until a head or three tails occur. If X denotes the number of tosses of the coin, then the mean of $X$ is

A bag contains 6 white and 4 black balls. A die is rolled once and the number of balls equal to the number obtained on the die are drawn from the bag at random. The probability that all the balls drawn are white is:

Let $S=\{M=[a_{ij}],a_{ij}\in \{0,1,2\},1\le i,j\le 2\}$ be a sample space and $A=\{M\in S:M\text{ is invertible}\}$ be an event. Then $P(A)$ is equal to

Let $\Omega$ be the sample space and $\mathrm{A}\subseteq \Omega$ be an event. Given below are two statements:
(S1): If $\mathrm{P}(\mathrm{A})=0$, then $\mathrm{A}=\varphi$
(S2): If $\mathrm{P}(\mathrm{A})=1$, then $\mathrm{A}=\Omega$
Then

If an unbiased die, marked with $-2,-1,0,1,2,3$ on its faces, is thrown five times, then the probability that the product of the outcomes is positive, is:

Two dice are thrown independently. Let A be the event that the number appeared on the $1^{\text{st }}$ die is less than the number appeared on the $2^{\text{nd }}$ die, B be the event that the number appeared on the $1^{\text{st }}$ die is even and that on the second die …

Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable X denote the number of rotten apples. If $\mu$ and ${\sigma }^2$ represent mean and variance of $X$, respectively, then …

Let N be the sum of the numbers appeared when two fair dice are rolled and let the probability that $\mathrm{N}-2,\sqrt{3\mathrm{N}},\mathrm{N}+2$ are in geometric progression be $\frac{\mathrm{k}}{48}$. Then the value of k is

25% of the population are smokers. A smoker has 27 times more chances to develop lung cancer then a non-smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is $\frac{k}{10}$. Then the value of k is ______.

The urns A, B and C contain 4 red, 6 black; 5 red, 5 black and $\lambda$ red, 4 black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball drawn is red and the probability that it is drawn from urn C is 0.4 then the square of the length of th …

Let A be the event that the absolute difference between two randomly choosen real numbers in the sample space $[0,60]$ is less than or equal to a. If $\mathrm{P}(A)=\frac{11}{36}$, then a is equal to ______.

A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is p. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the sam …

A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is

In a binomial distribution $\mathrm{B}(\mathrm{n},\mathrm{p})$, the sum and product of the mean & variance are 5 and 6 respectively, then find $6(\mathrm{n}+\mathrm{p}-\mathrm{q})$ is equal to :-

Let M be the maximum value of the product of two positive integers when their sum is 66. Let the sample space $S=\left\{x\in Z:x(66-x)\ge \frac{5}{9}M\right\}$ and the event $A=\{x\in S:x$ is a multiple of 3$\}$. Then $\mathrm{P}(\mathrm{A})$ is equal to

Bag I contains 3 red, 4 black and 3 white balls and Bag II contains 2 red, 5 black and 2 white balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be black in colour. Then the probability, that the transferred ba …

The sum and product of the mean and variance of a binomial distribution are 82.5 and 1350 respectively. Then the number of trials in the binomial distribution is:

Out of $60\%$ female and $40\%$ male candidates appearing in an exam, $60\%$ candidates qualify it. The number of females qualifying the exam is three times the number of males qualifying it. A candidate is randomly chosen from the qualified candidates. The probability, …

Let A and B be two events such that $\mathrm{P}(\mathrm{B}\mid \mathrm{A})=\frac{2}{5}$, $\mathrm{P}(\mathrm{A}\mid \mathrm{B})=\frac{1}{7}$ and $\mathrm{P}(\mathrm{A}\cap \mathrm{B})=\frac{1}{9}$. Consider
(S1) …

A bag contains 4 white and 6 black balls. Three balls are drawn at random from the bag. Let X be the number of white balls, among the drawn balls. If ${\sigma }^2$ is the variance of X , then $100{\sigma }^2$ is equal to

Let S be the sample space of all five digit numbers. If $p$ is the probability that a randomly selected number from S, is a multiple of 7 but not divisible by 5, then $9p$ is equal to

If the sum and the product of mean and variance of a binomial distribution are 24 and 128 respectively, then the probability of one or two successes is:

Let X be a binomially distributed random variable with mean 4 and variance $\frac{4}{3}$. Then $54P(X\le 2)$ is equal to

Let X have a binomial distribution $B(n,p)$ such that the sum and the product of the mean and variance of X are 24 and 128 respectively. If $P(X>n-3)=\frac{k}{2^n}$, then k is equal to

A six faced die is biased such that $3\times P(\text{a prime number})=6\times P(\text{a composite number})=2\times P(1)$. Let X be a random variable that counts the number of times one gets a perfect square on some throws of this die. If the die is thrown twice, then the m …

If A and B are two events such that $\mathrm{P}(\mathrm{A})=\frac{1}{3},\mathrm{P}(\mathrm{B})=\frac{1}{5}$ and $\mathrm{P}(\mathrm{A}\cup \mathrm{B})=\frac{1}{2}$, then …

The mean and variance of a binomial distribution are $\alpha$ and $\frac{\alpha }{3}$ respectively. If $P(X=1)=\frac{4}{243}$, then $P(X=4$ or $5)$ is equal to :

Let $E_1,E_2,E_3$ be three mutually exclusive events such that $P(E_1)=\frac{2+3p}{6},P(E_2)=\frac{2-p}{8}$ and $P(E_3)=\frac{1-p}{2}$. If the maximum and minimum values of p are $p_1$ and $p_2$, then $(p_1+p_2)$ is equal to :