# Probability · Mathematics · JEE Advanced

Start Practice## 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 $$1/35$$