# Probability · Mathematics · JEE Advanced

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JEE Advanced 2023 Paper 2 Online
Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulti...
JEE Advanced 2023 Paper 1 Online
Let $X=\left\{(x, y) \in \mathbb{Z} \times \mathbb{Z}: \frac{x^2}{8}+\frac{y^2}{20}... JEE Advanced 2022 Paper 2 Online Suppose that Box-I contains 8 red, 3 blue and 5 green balls, Box-II contains 24 red, 9 blue and 15 green balls, Box-III contains 1 blue, 12 green and... JEE Advanced 2022 Paper 1 Online Two players, $$P_{1}$$ and $$P_{2}$$, play a game against each other. In every round of the game, each player rolls a fair die once, where the six fac... JEE Advanced 2021 Paper 1 Online Consider three sets E1 = {1, 2, 3}, F1 = {1, 3, 4} and G1 = {2, 3, 4, 5}. Two elements are chosen at random, without replacement, from the set E1, and... JEE Advanced 2020 Paper 1 Offline Let C1 and C2 be two biased coins such that the probabilities of getting head in a single toss are $${{2 \over 3}}$$ and $${{1 \over 3}}$$, respective... JEE Advanced 2018 Paper 1 Offline There are five students S1, S2, S3, S4 and S5 in a music class and for them there are five seats R1, R2, R3, R4 and R5 arranged in a row, where initia... JEE Advanced 2018 Paper 1 Offline There are five students S1, S2, S3, S4 and S5 in a music class and for them there are five seats R1, R2, R3, R4 and R5 arranged in a row, where initia... JEE Advanced 2017 Paper 2 Offline Three randomly chosen nonnegative integers x, y and z are found to satisfy the equation x + y + z = 10. Then the probability that z is even, is JEE Advanced 2016 Paper 2 Offline Football teams $${T_1}$$ and $${T_2}$$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. Th... JEE Advanced 2016 Paper 2 Offline Football teams $${T_1}$$ and $${T_2}$$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. Th... JEE Advanced 2016 Paper 1 Offline A computer producing factory has only two plants $${T_1}$$ and $${T_2}.$$ Plant $${T_1}$$ produces $$20$$% and plant $${T_2}$$ produces $$80$$% of the... JEE Advanced 2014 Paper 2 Offline Three boys and two girls stand in a queue. The probability, that the number of boys ahead of every girl is at least one more than the number of girls... JEE Advanced 2014 Paper 2 Offline Box $$1$$ contains three cards bearing numbers $$1,2,3;$$ box $$2$$ contains five cards bearing numbers $$1,2,3,4,5;$$ and box $$3$$ contains seven ca... JEE Advanced 2014 Paper 2 Offline Box $$1$$ contains three cards bearing numbers $$1,2,3;$$ box $$2$$ contains five cards bearing numbers $$1,2,3,4,5;$$ and box $$3$$ contains seven ca... JEE Advanced 2013 Paper 2 Offline A box $${B_1}$$ contains $$1$$ white ball, $$3$$ red balls and $$2$$ black balls. Another box $${B_2}$$ contains $$2$$ white balls, $$3$$ red balls an... JEE Advanced 2013 Paper 2 Offline A box $${B_1}$$ contains $$1$$ white ball, $$3$$ red balls and $$2$$ black balls. Another box $${B_2}$$ contains $$2$$ white balls, $$3$$ red balls an... JEE Advanced 2013 Paper 1 Offline Four persons independently solve a certain problem correctly with probabilities $${1 \over 2},{3 \over 4},{1 \over 4},{1 \over 8}.$$ Then the probabil... IIT-JEE 2012 Paper 2 Offline Four fair dice $${D_1,}$$ $${D_2,}$$ $${D_3}$$ and $${D_4}$$ ; each having six faces numbered $$1, 2, 3, 4, 5$$ and $$6$$ are rolled simultaneously. T... IIT-JEE 2011 Paper 1 Offline The probability of the drawn ball from $${U_2}$$ being white is IIT-JEE 2011 Paper 1 Offline Given that the drawn ball from $${U_2}$$ is white, the probability that head appeared on the coin is IIT-JEE 2010 Paper 1 Offline Let $$\omega$$ be a complex cube root of unity with $$\omega \ne 1.$$ A fair die is thrown three times. If $${r_1},$$ $${r_2}$$ and $${r_3}$$ are th... IIT-JEE 2010 Paper 2 Offline A signal which can be green or red with probability $${4 \over 5}$$ and $${1 \over 5}$$ respectively, is received by station A and then transmitted to... IIT-JEE 2009 Paper 1 Offline The probability that X = 3 equals IIT-JEE 2009 Paper 1 Offline The conditional probability that $$X\ge6$$ given $$X>3$$ equals : IIT-JEE 2009 Paper 1 Offline The probability that $$X\ge3$$ equals : IIT-JEE 2008 Paper 2 Offline An experiment has 10 equally likely outcomes. Let A and B be two non-empty events of the experiment. If A consists of 4 outcomes, the number of outcom... IIT-JEE 2008 Paper 1 Offline Consider the system of equations $$ax+by=0; cx+dy=0,$$ where $$a,b,c,d$$ $$\in \left\{ {0,1} \right\}$$ STATEMENT - 1 : The probability that the sys... IIT-JEE 2007 One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that the Indian m... IIT-JEE 2007 Let $${H_1},{H_2},....,{H_n}$$ be mutually exclusive and exhaustive events with $$P\left( {{H_1}} \right) > 0,i = 1,2,.....,n.$$ Let $$E$$ be any o... IIT-JEE 2007 Paper 2 Offline Let $${E^c}$$ denote the complement of an event $$E.$$ Let $$E, F, G$$ be pairwise independent events with $$P\left( G \right) > 0$$ and $$P\left( ... IIT-JEE 2006 There are$$n$$urns, each of these contain$$n+1$$balls. The ith urn contains$$i$$white balls and$$(n+1-i)$$red balls. Let$${u_i}$$be the even... IIT-JEE 2006 There are$$n$$urns, each of these contain$$n+1$$balls. The ith urn contains$$i$$white balls and$$(n+1-i)$$red balls. Let$${u_i}$$be the even... IIT-JEE 2006 There are$$n$$urns, each of these contain$$n+1$$balls. The ith urn contains$$i$$white balls and$$(n+1-i)$$red balls. Let$${u_i}$$be the even... IIT-JEE 2005 Screening A six faced fair dice is thrown until$$1$$comes, then the probability that$$1$$comes in even no. of trials is IIT-JEE 2004 Screening If three distinct numbers are chosen randomly from the first$$100$$natural numbers, then the probability that all three of them are divisible by bot... IIT-JEE 2003 Screening Two numbers are selected randomly from the set$$S = \left\{ {1,2,3,4,5,6} \right\}$$without replacement one by one. The probability that minimum of ... IIT-JEE 2003 Screening If$$P\left( B \right) = {3 \over 4},P\left( {A \cap B \cap \overline C } \right) = {1 \over 3}$$and$$P\left( {\overline A \cap B \cap \overline C... IIT-JEE 1999 If the integers $$m$$ and $$n$$ are chosen at random from $$1$$ to $$100$$, then the probability that a number of the form $${7^m} + {7^n}$$ is divisi... IIT-JEE 1998 If from each of the three boxes containing $$3$$ white and $$1$$ black, $$2$$ white and $$2$$ black, $$1$$ white and $$3$$ black balls, one ball is dr... IIT-JEE 1998 There are four machines and it is known that exactly two of them are faulty. They are tested, one by one, in a random order till both the faulty machi... IIT-JEE 1998 If $$E$$ and $$F$$ are events with $$P\left( E \right) \le P\left( F \right)$$ and $$P\left( {E \cap F} \right) > 0,$$ then IIT-JEE 1998 Seven white balls and three black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals IIT-JEE 1998 A fair coin is tossed repeatedly. If the tail appears on first four tosses, then the probability of the head appearing on the fifth toss equals IIT-JEE 1996 For the three events $$A, B,$$ and $$C,P$$ (exactly one of the events $$A$$ or $$B$$ occurs) $$=P$$ (exactly one of the two events $$B$$ or $$C$$ occu... IIT-JEE 1995 Screening The probability of India winning a test match against West Indies is $$1/2$$. Assuming independence from match to match the probability that in a $$5... IIT-JEE 1995 Screening Three of six vertices of a regular hexagon are chosen at random. The probability that the triangle with three vertices is equilateral, equals IIT-JEE 1994 Let$$A, B, C$$be three mutually independent events. Consider the two statements$${S_1}$$and$${S_2}{S_1}\,:\,A$$and$$B \cup C$$are indepen... IIT-JEE 1993 An unbiased die with faces marked$$1,2,3,4,5$$and$$6$$is rolled four times. Out of four face values obtained, the probability that the minimum fac... IIT-JEE 1992 India plays two matches each with West Indies and Australia. In any match the probabilities of India getting, points$$0,1$$and$$2$$are$$0.45... IIT-JEE 1988 One hundred identical coins, each with probability, $$p,$$ of showing up heads are tossed once. If $$0 < p < 1$$ and the probability of heads sh... IIT-JEE 1986 A student appears for tests, $$I$$, $$II$$ and $$III$$. The student is successful if he passes either in tests $$I$$ and $$II$$ or tests $$I$$ and $$I... IIT-JEE 1986 The probability that at least one of the events$$A$$and$$B$$occurs is$$0.6$$. If$$A$$and$$B$$occur simultaneously with probability$$0.2,$$t... IIT-JEE 1984 A box contains$$24$$identical balls of which$$12$$are white and$$12$$are black. The balls are drawn at random from the box one at a time with re... IIT-JEE 1984 Three identical dice are rolled. The probability that the same number will appear on each of them is IIT-JEE 1983 Fifteen coupons are numbered$$1, 2 ........15,$$respectively. Seven coupons are selected at random one at a time with replacement. The probability t... IIT-JEE 1982 If$$A$$and$$B$$are two events such that$$P\left( A \right) > 0,$$and$$P\left( B \right) \ne 1,$$then$$P\left( {{{\overline A } \over {\ove... IIT-JEE 1980 The probability that an event $$A$$ happens in one trial of an experiment is $$0.4.$$ Three independent trials of the experiment are performed. The pr... IIT-JEE 1980 Two events $$A$$ and $$B$$ have probabilities $$0.25$$ and $$0.50$$ respectively. The probability that both $$A$$ and $$B$$ occur simultaneously is $$... IIT-JEE 1979 Two fair dice are tossed. Let$$x$$be the event that the first die shows an even number and$$y$$be the event that the second die shows an odd numbe... ## Numerical JEE Advanced 2023 Paper 2 Online Let X be the set of all five digit numbers formed using 1,2,2,2,4,4,0. For example, 22240 is in X while 02244 and 44422 are not in X. Suppose th... JEE Advanced 2023 Paper 2 Online Let p_i be the probability that a randomly chosen point has i many friends, i=0,1,2,3,4. Let X be a random variable such that for i=0,1,2,3,4... JEE Advanced 2023 Paper 2 Online Two distinct points are chosen randomly out of the points A_1, A_2, \ldots, A_{49}. Let p be the probability that they are friends. Then the value... JEE Advanced 2022 Paper 1 Online In a study about a pandemic, data of 900 persons was collected. It was found that 190 persons had symptom of fever, 220 persons had symptom of cough... JEE Advanced 2021 Paper 2 Online A number of chosen at random from the set {1, 2, 3, ....., 2000}. Let p be the probability that the chosen number is a multiple of 3 or a multiple of ... JEE Advanced 2021 Paper 1 Online Three numbers are chosen at random, one after another with replacement, from the set S = {1, 2, 3, ......, 100}. Let p1 be the probability that the ma... JEE Advanced 2021 Paper 1 Online Three numbers are chosen at random, one after another with replacement, from the set S = {1, 2, 3, ......, 100}. Let p1 be the probability that the ma... JEE Advanced 2020 Paper 2 Offline The probability that a missile hits a target successfully is 0.75. In order to destroy the target completely, at least three successful hits are requi... JEE Advanced 2020 Paper 2 Offline Two fair dice, each with faces numbered 1, 2, 3, 4, 5 and 6, are rolled together and the sum of the numbers on the faces is observed. This process is ... JEE Advanced 2019 Paper 1 Offline Let S be the sample space of all 3$$ \times $$3 matrices with entries from the set {0, 1}. Let the events E1 and E2 be given byE1 = {A$$ \in $$S : d... JEE Advanced 2015 Paper 1 Offline The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least$$0.96,$$is JEE Advanced 2013 Paper 1 Offline Of the three independent events$${E_1},{E_2}$$and$${E_3},$$the probability that only$${E_1}$$occurs is$$\alpha ,$$only$${E_2}$$occurs is$$\... ## MCQ (More than One Correct Answer) JEE Advanced 2021 Paper 1 Online Let E, F and G be three events having probabilities $$P(E) = {1 \over 8}$$, $$P(F) = {1 \over 6}$$ and $$P(G) = {1 \over 4}$$, and let P (E $$\cap$$ F... JEE Advanced 2019 Paper 1 Offline There are three bags B1, B2 and B3. The bag B1 contains 5 red and 5 green balls, B2 contains 3 red and 5 green balls, and B3 contains 5 red and 3 gree... JEE Advanced 2017 Paper 1 Offline Let X and Y be two events such that $$P(X) = {1 \over 3}$$, $$P(X|Y) = {1 \over 2}$$ and $$P(Y|X) = {2 \over 5}$$. Then JEE Advanced 2015 Paper 2 Offline Let $${n_1}$$ and $${n_2}$$ be the number of red and black balls, respectively, in box $${\rm I}$$. Let $${n_3}$$ and $${n_4}$$ be the number of red ... JEE Advanced 2015 Paper 2 Offline Let $${n_1}$$ and $${n_2}$$ be the number of red and black balls, respectively, in box $${\rm I}$$. Let $${n_3}$$ and $${n_4}$$ be the number of red ... IIT-JEE 2012 Paper 2 Offline Let $$X$$ and $$Y$$ be two events such that $$P\left( {X|Y} \right) = {1 \over 2},$$ $$P\left( {Y|X} \right) = {1 \over 3}$$ and $$P\left( {X \cap Y} ... IIT-JEE 2012 Paper 1 Offline A ship is fitted with three engines$${E_1},{E_2}$$and$${E_3}$$. The engines function independently of each other with respective probabilities$${1... IIT-JEE 2011 Paper 2 Offline Let $$E$$ and $$F$$ be two independent events. The probability that exactly one of them occurs is $$\,{{11} \over {25}}$$ and the probability of none ... IIT-JEE 1999 The probabilities that a student passes in Mathematics, Physics and Chemistry are $$m, p$$ and $$c,$$ respectively. Of these subjects, the student has... IIT-JEE 1998 If $$\overline E$$ and $$\overline F$$ are the complementary events of events $$E$$ and $$F$$ respectively and if $$0 < P\left( F \right) < 1... IIT-JEE 1995 Screening Let$$0 < P\left( A \right) < 1,0 < P\left( B \right) < 1$$and$$P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\... IIT-JEE 1993 $$E$$ and $$F$$ are two independent events. The probability that both $$E$$ and $$F$$ happen is $$1/12$$ and the probability that neither $$E$$ nor $$... IIT-JEE 1991 For any two events$$A$$and$$B$$in a simple space IIT-JEE 1989 If$$E$$and$$F$$are independent events such that$$0 < P\left( E \right) < 1$$and$$0 < P\left( F \right) < 1,$$then IIT-JEE 1988 For two given events$$A$$and$$B,P\left( {A \cap B} \right)$$IIT-JEE 1984 If$$M$$and$$N$$are any two events, the probability that exactly one of them occurs is ## Subjective IIT-JEE 2005 A person goes to office either by car, scooter, bus or train, the probability of which being$${1 \over 7},{3 \over 7},{2 \over 7}$$and$${1 \over 7}... IIT-JEE 2004 $$A$$ and $$B$$ are two independent events. $$C$$ is even in which exactly one of $$A$$ or $$B$$ occurs. Prove that $$P\left( C \right) \ge P\left( {A... IIT-JEE 2004 A box contains$$12$$red and$$6$$white balls. Balls are drawn from the box one at a time without replacement. If in$$6$$draws there are at least ... IIT-JEE 2003$$A$$is targeting to$$B, B$$and$$C$$are targeting to$$A.$$Probability of hitting the target by$$A,B$$and$$C$$are$${2 \over 3},{1 \over 2}... IIT-JEE 2003 For a student to qualify, he must pass at least two out of three exams. The probability that he will pass the 1st exam is $$p.$$ If he fails in one of... IIT-JEE 2002 A box contains $$N$$ coins, $$m$$ of which are fair and the rest are biased. The probability of getting a head when a fair coin is tossed is $$1/2$$, ... IIT-JEE 2001 An urn contains $$m$$ white and $$n$$ black balls. A ball is drawn at random and is put back into the urn along with $$k$$ additional balls of the sam... IIT-JEE 2001 An unbiased die, with faces numbered $$1,2,3,4,5,6,$$ is thrown $$n$$ times and the list of $$n$$ numbers showing up is noted. What is the probability... IIT-JEE 2000 A coin has probability $$p$$ of showing head when tossed. It is tossed $$n$$ times. Let $${p_n}$$ denote the probability that no two (or more) consecu... IIT-JEE 1999 Eight players $${P_1},{P_2},.....{P_8}$$ play a knock-out tournament. It is known that whenever the players $${P_i}$$ and $${P_j}$$ play, the player$...
IIT-JEE 1998
Three players, $$A,B$$ and $$C,$$ toss a coin cyclically in that order (that is $$A, B, C, A, B, C, A, B,...$$) till a head shows. Let $$p$$ be the pr...
IIT-JEE 1998
Let $${C_1}$$ and $${C_2}$$ be the graphs of the functions $$y = {x^2}$$ and $$y = 2x,$$ $$0 \le x \le 1$$ respectively. Let $${C_3}$$ be the graph of...
IIT-JEE 1997
If $$p$$ and $$q$$ are chosen randomly from the set $$\left\{ {1,2,3,4,5,6,7,8,9,10} \right\},$$ with replacement, determine the probability that the ...
IIT-JEE 1996
In how many ways three girls and nine boys can be seated in two vans, each having numbered seats, $$3$$ in the front and $$4$$ at the back? How many ...
IIT-JEE 1994
An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the number obtained by adding the numbers on the two faces ...
IIT-JEE 1993
Numbers are selected at random, one at a time, from the two- digit numbers $$00, 01, 02 ......, 99$$ with replacement. An event $$E$$ occurs if only i...
IIT-JEE 1992
A lot contains $$50$$ defective and $$50$$ non defective bulbs. Two bulbs are drawn at random, one at a time, with replacement. The events $$A, B, C$$...
IIT-JEE 1991
In a test an examine either guesses or copies or knows the answer to a multiple choice question with four choices. The probability that he make a gues...
IIT-JEE 1990
A is a set containing $$n$$ elements. $$A$$ subset $$P$$ of $$A$$ is chosen at random. The set $$A$$ is reconstructed by replacing the elements of $$P... IIT-JEE 1989 Suppose the probability for A to win a game against B is$$0.4.$$If$$A$$has an option of playing either a "best of$$3$$games" or a "best of$$5$$... IIT-JEE 1988 A box contains$$2$$fifty paise coins,$$5$$twenty five paise coins and a certain fixed number$$N\,\,\left( { \ge 2} \right)$$of ten and five pais... IIT-JEE 1987 A man takes a step forward with probability$$0.4$$and backwards with probability$$0.6$$Find the probability that at the end of eleven steps he is ... IIT-JEE 1986 A lot contains$$20$$articles. The probability that the lot contains exactly$$2$$defective articles is$$0.4$$and the probability that the lot con... IIT-JEE 1985 In a multiple-choice question there are four alternative answers, of which one or more are correct. A candidate will get marks in the question only if... IIT-JEE 1984 In a certain city only two newspapers$$A$$and$$B$$are published, it is known that$$25$$% of the city population reads$$A$$and$$20$$% reads$$B...
IIT-JEE 1983
Cards are drawn one by one at random from a well - shuffled full pack of $$52$$ playing cards until $$2$$ aces are obtained for the first time. If $$N... IIT-JEE 1983$$A, B, C$$are events such that$$P\left( A \right) = 0.3,P\left( B \right) = 0.4,P\left( C \right) = 0.8P\left( {AB} \right) = 0.08,P\left( {AC...
IIT-JEE 1982
$$A$$ and $$B$$ are two candidates seeking admission in $$IIT.$$ The probability that $$A$$ is selected is $$0.5$$ and the probability that both $$A$$...
IIT-JEE 1981
An anti-aircraft gun can take a maximum of four shots at an enemy plane moving away from it. The probabilities of hitting the plane at the first, se...
IIT-JEE 1979
Six boys and six girls sit in a row randomly. Find the probability that (i) the six girls sit together (ii) the boys and girls sit alternately....
IIT-JEE 1978
Balls are drawn one-by-one without replacement from a box containing $$2$$ black, $$4$$ white and $$3$$ red balls till all the balls are drawn. Find t...

IIT-JEE 1994
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