Probability · Mathematics · JEE Advanced

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MCQ (Single Correct Answer)

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 $$P\left( A \right) = 0.3,P\left( B \right) = 0.4,P\left( C \right) = 0.8$$ $$P\left( {AB} \right) = 0.08,P\left( {AC...
IIT-JEE 1982
$$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...

Fill in the Blanks

IIT-JEE 1994
If two events $$A$$ and $$B$$ are such that $$P\,\,\left( {{A^c}} \right)\,\, = \,\,0.3,\,\,P\left( B \right) = 0.4$$ and $$P\left( {A \cap {B^c}} \ri...
IIT-JEE 1992
Three faces of a fair die are yellow, two faces red and one blue. The die is tossed three times. The probability that the colours, yellow, red and bl...
IIT-JEE 1991
If the mean and the variance of binomial variate $$X$$ are $$2$$ and $$1$$ respectively, then the probability that $$X$$ takes a value greater than on...
IIT-JEE 1990
Let $$A$$ and $$B$$ be two events such that $$P\,\,\left( A \right)\,\, = \,\,0.3$$ and $$P\left( {A \cup B} \right) = 0.8.$$ If $$A$$ and $$B$$ are i...
IIT-JEE 1989
A pair of fair dice is rolled together till a sum of either $$5$$ or $$7$$ is obtained. Then the probability that $$5$$ comes before $$7$$ is ...........
IIT-JEE 1988
Urn $$A$$ contains $$6$$ red and $$4$$ black balls and urn $$B$$ contains $$4$$ red and $$6$$ black balls. One ball is drawn at random from urn $$A$$ ...
IIT-JEE 1986
If $${{1 + 3p} \over 3},\,\,\,{{1 - p} \over 4}$$ and $$\,{{1 - 2p} \over 2}$$ are the probabilities of three mutually exclusive events, then the set ...
IIT-JEE 1985
A box contains $$100$$ tickets numbered $$1, 2, ....., 100.$$ Two tickets are chosen at random. It is given that the maximum number on the two chosen ...
IIT-JEE 1985
$$P\left( {A \cup B} \right) = P\left( {A \cap B} \right)$$ if and only if the relation between $$P(A)$$ and $$P(B)$$ is .............
IIT-JEE 1981
For a biased die the probabilities for the different faces to turn up are given below : This die tossed and you are told that either face $$1$$ or f...

True or False

IIT-JEE 1989
If the probability for $$A$$ to fail in an examination is $$0.2$$ and that for $$B$$ is $$0.3$$, then the probability that either $$A$$ or $$B$$ fails...
IIT-JEE 1983
If the letters of the word "Assassin" are written down at random in a row, the probability that no two S's occur together is $$1/35$$
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