Probability — JEE Main PYQs

A random variable $X$ takes values $0,1,2,3$ with probabilities $\frac{2a+1}{30}$, $\frac{8a-1}{30}$, $\frac{4a+1}{30}$, b respectively, where $a,b\in \mathbf{R}$. Let $\mu$ and $\sigma$ respectively be the mean and standard deviation of X such that ${\sigma }^2+{\mu }^2=2$.
Then …

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

If a random variable $x$ has the probability distribution

then …

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

Bag A contains 9 white and 8 black balls, while bag B contains 6 white and 4 black balls. One ball is randomly picked up from the bag B and mixed up with the balls in the bag A. Then a ball is randomly drawn from the bag A. If the probability, that the ball drawn is white, 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 $\mathrm{P}(\mathrm{S})$ denote the power set of S . Let E be an event of choosing an ordered pair (A, B) from the set $\mathrm{P}(\mathrm{S})\times \mathrm{P}(\mathrm{S})$ such that $\mathrm{A}\cap \mathrm{B}=\varnothing$. If the probabil …

A bag contains 10 balls out of which k are red and $(10-\mathrm{k})$ are black, where $0\le \mathrm{k}\le 10$. If three balls are drawn at random without replacement and all of them are found to be black, then the probability that the bag contains 1 red and 9 black balls …

The probability distribution of a random variable X is given below :
| X | 4 k | $\frac{30}{7}\mathrm{k}$ | $\frac{32}{7}\mathrm{k}$ | $\frac{34}{7}\mathrm{k}$ | $\frac{36}{7}\mathrm{k}$ | $\frac{38}{7}\mathrm{k}$ | $\frac{40}{7}\mathrm{k}$ | 6 k || P(X) | …

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 $\frac{\mathrm{a}}{\mathrm{b}}$, where $\mathrm{a},\mathrm{b}\in \mathrm{N}$ and $\gcd (\mathrm{a},\mathrm{b})=1$, then …

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 ____.

A letter is known to have arrived by post either from KANPUR or from ANANTPUR. On the 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 …

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 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 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 …

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 $\frac{\mathrm{m}}{\mathrm{n}},\gcd (\mathrm{m},\mathrm{n})=1$, then …

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 $64\left(\mu +{\sigma }^2\right)$ 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 $\frac{\mathrm{m}}{\mathrm{n}}$, where …

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

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 throw …

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 :

Let $\mathrm{A}=\left[{\mathrm{a}}_{\mathrm{ij}}\right]$ be a $2\times 2$ matrix such that ${\mathrm{a}}_{\mathrm{ij}}\in \{0,1\}$ for all i and j . Let the random variable X denote the possible values of the determinant of the matrix A . Then, the variance of X 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 equa …

Bag $B_1$ contains 6 white and 4 blue balls, Bag $B_2$ contains 4 white and 6 blue balls, and $\mathrm{Bag}{\mathrm{B}}_3$ 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 t …

Two number ${\mathrm{k}}_1$ and ${\mathrm{k}}_2$ 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 $\mathrm{A}=\left[{\mathrm{a}}_{\mathrm{ij}}\right]$ 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 $\mathrm{P}(\mathrm{E})$ 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

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

A person chooses a bag at random and takes out a ball. If the ball i …

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

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

A card from a pack of 52 cards is lost. From the remaining 51 cards, n cards are drawn and are found to be spades. If the probability of the lost card to be a spade is $\frac{11}{50}$, the n 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 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 $\frac{\mathrm{m}}{\mathrm{n}},\gcd (\mathrm{m},\mathrm{n})=1$, then …

Let a random variable X take values $0,1,2,3$ with $\mathrm{P}(\mathrm{X}=0)=\mathrm{P}(\mathrm{X}=1)=\mathrm{p},\mathrm{P}(\mathrm{X}=2)=\mathrm{P}(\mathrm{X}=3)$ and $\mathrm{E}\left({\mathrm{X}}^2\right)=2\mathrm{E}(\mathrm{X})$. Then the value of $8\mathrm{p}-1$ is …

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

Three defective oranges are accidentally mixed with seven good ones and it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If x denotes the number of defective oranges, then the variance of x 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 random variable $X$: $X:0,2,4,6,8$; $P(X):a,2a,a+b,2b,3b$ is $\frac{46}{9}$, then the variance of the distribution is

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as $\frac{1}{3}$ and $\frac{2}{3}$ respectively. Let $x$ be the number of matches that the team wins, and $y$ be the number of matches that they lose. If the probability …

The coefficients $a,b,c$, in the quadratic equation $a{x}^2+bx+c=0$ 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 $a{x}^2+bx+c=0$ 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 :

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 p …

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 $\frac{m}{n}$ where …

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 positive 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 f …