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JEE Advanced

Probability

Mathematics

Previous Years Questions

Numerical

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A number of chosen at random from the set {1, 2, 3, ....., 2000}. Let p be the probability that the chosen number is a m...
JEE Advanced 2021 Paper 2 Online
Three numbers are chosen at random, one after another with replacement, from the set S = {1, 2, 3, ......, 100}. Let p1 ...
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 ...
JEE Advanced 2021 Paper 1 Online
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...
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 t...
JEE Advanced 2020 Paper 2 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...
JEE Advanced 2019 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 ...
JEE Advanced 2015 Paper 1 Offline
Of the three independent events $${E_1},{E_2}$$ and $${E_3},$$ the probability that only $${E_1}$$ occurs is $$\alpha ,$...
JEE Advanced 2013 Paper 1 Offline

MCQ (More than One Correct Answer)

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Let E, F and G be three events having probabilities $$P(E) = {1 \over 8}$$, $$P(F) = {1 \over 6}$$ and $$P(G) = {1 \over...
JEE Advanced 2021 Paper 1 Online
There are three bags B1, B2 and B3. The bag B1 contains 5 red and 5 green balls, B2 contains 3 red and 5 green balls, an...
JEE Advanced 2019 Paper 1 Offline
Let X and Y be two events such that $$P(X) = {1 \over 3}$$, $$P(X|Y) = {1 \over 2}$$ and $$P(Y|X) = {2 \over 5}$$. Then
JEE Advanced 2017 Paper 1 Offline
Let $${n_1}$$ and $${n_2}$$ be the number of red and black balls, respectively, in box $${\rm I}$$. Let $${n_3}$$ and $...
JEE Advanced 2015 Paper 2 Offline
Let $${n_1}$$ and $${n_2}$$ be the number of red and black balls, respectively, in box $${\rm I}$$. Let $${n_3}$$ and $...
JEE Advanced 2015 Paper 2 Offline
Let $$X$$ and $$Y$$ be two events such that $$P\left( {X|Y} \right) = {1 \over 2},$$ $$P\left( {Y|X} \right) = {1 \over ...
IIT-JEE 2012 Paper 2 Offline
A ship is fitted with three engines $${E_1},{E_2}$$ and $${E_3}$$. The engines function independently of each other with...
IIT-JEE 2012 Paper 1 Offline
Let $$E$$ and $$F$$ be two independent events. The probability that exactly one of them occurs is $$\,{{11} \over {25}}$...
IIT-JEE 2011 Paper 2 Offline
The probabilities that a student passes in Mathematics, Physics and Chemistry are $$m, p$$ and $$c,$$ respectively. Of t...
IIT-JEE 1999
If $$\overline E $$ and $$\overline F $$ are the complementary events of events $$E$$ and $$F$$ respectively and if $$0...
IIT-JEE 1998
Let $$0 < P\left( A \right) < 1,0 < P\left( B \right) < 1$$ and $$P\left( {A \cup B} \right) = P\left( A \r...
IIT-JEE 1995 Screening
$$E$$ and $$F$$ are two independent events. The probability that both $$E$$ and $$F$$ happen is $$1/12$$ and the probabi...
IIT-JEE 1993
For any two events $$A$$ and $$B$$ in a simple space
IIT-JEE 1991
If $$E$$ and $$F$$ are independent events such that $$0 < P\left( E \right) < 1$$ and $$0 < P\left( F \right) &...
IIT-JEE 1989
For two given events $$A$$ and $$B,$$ $$P\left( {A \cap B} \right)$$
IIT-JEE 1988
If $$M$$ and $$N$$ are any two events, the probability that exactly one of them occurs is
IIT-JEE 1984

MCQ (Single Correct Answer)

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Consider three sets E1 = {1, 2, 3}, F1 = {1, 3, 4} and G1 = {2, 3, 4, 5}. Two elements are chosen at random, without rep...
JEE Advanced 2021 Paper 1 Online
Let C1 and C2 be two biased coins such that the probabilities of getting head in a single toss are $${{2 \over 3}}$$ and...
JEE Advanced 2020 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 a...
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 a...
JEE Advanced 2018 Paper 1 Offline
Three randomly chosen nonnegative integers x, y and z are found to satisfy the equation x + y + z = 10. Then the probabi...
JEE Advanced 2017 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...
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...
JEE Advanced 2016 Paper 2 Offline
A computer producing factory has only two plants $${T_1}$$ and $${T_2}.$$ Plant $${T_1}$$ produces $$20$$% and plant $${...
JEE Advanced 2016 Paper 1 Offline
Box $$1$$ contains three cards bearing numbers $$1,2,3;$$ box $$2$$ contains five cards bearing numbers $$1,2,3,4,5;$$ a...
JEE Advanced 2014 Paper 2 Offline
Box $$1$$ contains three cards bearing numbers $$1,2,3;$$ box $$2$$ contains five cards bearing numbers $$1,2,3,4,5;$$ a...
JEE Advanced 2014 Paper 2 Offline
Three boys and two girls stand in a queue. The probability, that the number of boys ahead of every girl is at least one...
JEE Advanced 2014 Paper 2 Offline
A box $${B_1}$$ contains $$1$$ white ball, $$3$$ red balls and $$2$$ black balls. Another box $${B_2}$$ contains $$2$$ w...
JEE Advanced 2013 Paper 2 Offline
A box $${B_1}$$ contains $$1$$ white ball, $$3$$ red balls and $$2$$ black balls. Another box $${B_2}$$ contains $$2$$ w...
JEE Advanced 2013 Paper 2 Offline
Four persons independently solve a certain problem correctly with probabilities $${1 \over 2},{3 \over 4},{1 \over 4},{1...
JEE Advanced 2013 Paper 1 Offline
Four fair dice $${D_1,}$$ $${D_2,}$$ $${D_3}$$ and $${D_4}$$ ; each having six faces numbered $$1, 2, 3, 4, 5$$ and $$6$...
IIT-JEE 2012 Paper 2 Offline
Given that the drawn ball from $${U_2}$$ is white, the probability that head appeared on the coin is
IIT-JEE 2011 Paper 1 Offline
The probability of the drawn ball from $${U_2}$$ being white is
IIT-JEE 2011 Paper 1 Offline
Let $$\omega $$ be a complex cube root of unity with $$\omega \ne 1.$$ A fair die is thrown three times. If $${r_1},$$ ...
IIT-JEE 2010 Paper 1 Offline
A signal which can be green or red with probability $${4 \over 5}$$ and $${1 \over 5}$$ respectively, is received by sta...
IIT-JEE 2010 Paper 2 Offline
A fair die is tossed repeatedly until a six is obtained. Let $$X$$ denote the number of tosses required. The conditiona...
IIT-JEE 2009
A fair die is tossed repeatedly until a six is obtained. Let $$X$$ denote the number of tosses required. The probabil...
IIT-JEE 2009
A fair die is tossed repeatedly until a six is obtained. Let $$X$$ denote the number of tosses required. The probabilit...
IIT-JEE 2009
Consider the system of equations $$ax+by=0; cx+dy=0,$$ where $$a,b,c,d$$ $$ \in \left\{ {0,1} \right\}$$ STATEMENT - 1 ...
IIT-JEE 2008
An experiment has $$10$$ equally likely outcomes. Let $$A$$ and $$B$$ be non-empty events of the experiment. If $$A$$ co...
IIT-JEE 2008
Let $${H_1},{H_2},....,{H_n}$$ be mutually exclusive and exhaustive events with $$P\left( {{H_1}} \right) > 0,i = 1,2...
IIT-JEE 2007
One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional...
IIT-JEE 2007
Let $${E^c}$$ denote the complement of an event $$E.$$ Let $$E, F, G$$ be pairwise independent events with $$P\left( G \...
IIT-JEE 2007
There are $$n$$ urns, each of these contain $$n+1$$ balls. The ith urn contains $$i$$ white balls and $$(n+1-i)$$ red ba...
IIT-JEE 2006
There are $$n$$ urns, each of these contain $$n+1$$ balls. The ith urn contains $$i$$ white balls and $$(n+1-i)$$ red ba...
IIT-JEE 2006
There are $$n$$ urns, each of these contain $$n+1$$ balls. The ith urn contains $$i$$ white balls and $$(n+1-i)$$ red ba...
IIT-JEE 2006
A six faced fair dice is thrown until $$1$$ comes, then the probability that $$1$$ comes in even no. of trials is
IIT-JEE 2005 Screening
If three distinct numbers are chosen randomly from the first $$100$$ natural numbers, then the probability that all thre...
IIT-JEE 2004 Screening
If $$P\left( B \right) = {3 \over 4},P\left( {A \cap B \cap \overline C } \right) = {1 \over 3}$$ and $$P\left( {\overl...
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. Th...
IIT-JEE 2003 Screening
If the integers $$m$$ and $$n$$ are chosen at random from $$1$$ to $$100$$, then the probability that a number of the fo...
IIT-JEE 1999
A fair coin is tossed repeatedly. If the tail appears on first four tosses, then the probability of the head appearing o...
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...
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 or...
IIT-JEE 1998
If $$E$$ and $$F$$ are events with $$P\left( E \right) \le P\left( F \right)$$ and $$P\left( {E \cap F} \right) > 0,$...
IIT-JEE 1998
If from each of the three boxes containing $$3$$ white and $$1$$ black, $$2$$ white and $$2$$ black, $$1$$ white and $$3...
IIT-JEE 1998
For the three events $$A, B,$$ and $$C,P$$ (exactly one of the events $$A$$ or $$B$$ occurs) $$=P$$ (exactly one of the ...
IIT-JEE 1996
The probability of India winning a test match against West Indies is $$1/2$$. Assuming independence from match to match...
IIT-JEE 1995 Screening
Three of six vertices of a regular hexagon are chosen at random. The probability that the triangle with three vertices i...
IIT-JEE 1995 Screening
Let $$A, B, C$$ be three mutually independent events. Consider the two statements $${S_1}$$ and $${S_2}$$ $${S_1}\,:\,A$...
IIT-JEE 1994
An unbiased die with faces marked $$1,2,3,4,5$$ and $$6$$ is rolled four times. Out of four face values obtained, the pr...
IIT-JEE 1993
India plays two matches each with West Indies and Australia. In any match the probabilities of India getting, points $$0...
IIT-JEE 1992
One hundred identical coins, each with probability, $$p,$$ of showing up heads are tossed once. If $$0 < p < 1$$ a...
IIT-JEE 1988
The probability that at least one of the events $$A$$ and $$B$$ occurs is $$0.6$$. If $$A$$ and $$B$$ occur simultaneous...
IIT-JEE 1986
A student appears for tests, $$I$$, $$II$$ and $$III$$. The student is successful if he passes either in tests $$I$$ and...
IIT-JEE 1986
A box contains $$24$$ identical balls of which $$12$$ are white and $$12$$ are black. The balls are drawn at random from...
IIT-JEE 1984
Three identical dice are rolled. The probability that the same number will appear on each of them is
IIT-JEE 1984
Fifteen coupons are numbered $$1, 2 ........15,$$ respectively. Seven coupons are selected at random one at a time with ...
IIT-JEE 1983
If $$A$$ and $$B$$ are two events such that $$P\left( A \right) > 0,$$ and $$P\left( B \right) \ne 1,$$ then $$P\left...
IIT-JEE 1982
Two events $$A$$ and $$B$$ have probabilities $$0.25$$ and $$0.50$$ respectively. The probability that both $$A$$ and $$...
IIT-JEE 1980
The probability that an event $$A$$ happens in one trial of an experiment is $$0.4.$$ Three independent trials of the ex...
IIT-JEE 1980
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...
IIT-JEE 1979

Subjective

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A person goes to office either by car, scooter, bus or train, the probability of which being $${1 \over 7},{3 \over 7},{...
IIT-JEE 2005
A box contains $$12$$ red and $$6$$ white balls. Balls are drawn from the box one at a time without replacement. If in $...
IIT-JEE 2004
$$A$$ and $$B$$ are two independent events. $$C$$ is even in which exactly one of $$A$$ or $$B$$ occurs. Prove that $$P\...
IIT-JEE 2004
For a student to qualify, he must pass at least two out of three exams. The probability that he will pass the 1st exam i...
IIT-JEE 2003
$$A$$ is targeting to $$B, B$$ and $$C$$ are targeting to $$A.$$ Probability of hitting the target by $$A,B$$ and $$C$$ ...
IIT-JEE 2003
A box contains $$N$$ coins, $$m$$ of which are fair and the rest are biased. The probability of getting a head when a fa...
IIT-JEE 2002
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 ...
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 $$k...
IIT-JEE 2001
A coin has probability $$p$$ of showing head when tossed. It is tossed $$n$$ times. Let $${p_n}$$ denote the probability...
IIT-JEE 2000
Eight players $${P_1},{P_2},.....{P_8}$$ play a knock-out tournament. It is known that whenever the players $${P_i}$$ an...
IIT-JEE 1999
Let $${C_1}$$ and $${C_2}$$ be the graphs of the functions $$y = {x^2}$$ and $$y = 2x,$$ $$0 \le x \le 1$$ respectively....
IIT-JEE 1998
Three players, $$A,B$$ and $$C,$$ toss a coin cyclically in that order (that is $$A, B, C, A, B, C, A, B,...$$) till a h...
IIT-JEE 1998
If $$p$$ and $$q$$ are chosen randomly from the set $$\left\{ {1,2,3,4,5,6,7,8,9,10} \right\},$$ with replacement, deter...
IIT-JEE 1997
In how many ways three girls and nine boys can be seated in two vans, each having numbered seats, $$3$$ in the front an...
IIT-JEE 1996
An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the number obtained by adding...
IIT-JEE 1994
Numbers are selected at random, one at a time, from the two- digit numbers $$00, 01, 02 ......, 99$$ with replacement. A...
IIT-JEE 1993
A lot contains $$50$$ defective and $$50$$ non defective bulbs. Two bulbs are drawn at random, one at a time, with repla...
IIT-JEE 1992
In a test an examine either guesses or copies or knows the answer to a multiple choice question with four choices. The p...
IIT-JEE 1991
A is a set containing $$n$$ elements. $$A$$ subset $$P$$ of $$A$$ is chosen at random. The set $$A$$ is reconstructed by...
IIT-JEE 1990
Suppose the probability for A to win a game against B is $$0.4.$$ If $$A$$ has an option of playing either a "best of $$...
IIT-JEE 1989
A box contains $$2$$ fifty paise coins, $$5$$ twenty five paise coins and a certain fixed number $$N\,\,\left( { \ge 2} ...
IIT-JEE 1988
A man takes a step forward with probability $$0.4$$ and backwards with probability $$0.6$$ Find the probability that at ...
IIT-JEE 1987
A lot contains $$20$$ articles. The probability that the lot contains exactly $$2$$ defective articles is $$0.4$$ and th...
IIT-JEE 1986
In a multiple-choice question there are four alternative answers, of which one or more are correct. A candidate will get...
IIT-JEE 1985
In a certain city only two newspapers $$A$$ and $$B$$ are published, it is known that $$25$$% of the city population rea...
IIT-JEE 1984
Cards are drawn one by one at random from a well - shuffled full pack of $$52$$ playing cards until $$2$$ aces are obtai...
IIT-JEE 1983
$$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( {...
IIT-JEE 1983
$$A$$ and $$B$$ are two candidates seeking admission in $$IIT.$$ The probability that $$A$$ is selected is $$0.5$$ and t...
IIT-JEE 1982
An anti-aircraft gun can take a maximum of four shots at an enemy plane moving away from it. The probabilities of hitti...
IIT-JEE 1981
Six boys and six girls sit in a row randomly. Find the probability that (i) the six girls sit together (ii) the boys a...
IIT-JEE 1979
Balls are drawn one-by-one without replacement from a box containing $$2$$ black, $$4$$ white and $$3$$ red balls till a...
IIT-JEE 1978

Fill in the Blanks

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If two events $$A$$ and $$B$$ are such that $$P\,\,\left( {{A^c}} \right)\,\, = \,\,0.3,\,\,P\left( B \right) = 0.4$$ an...
IIT-JEE 1994
Three faces of a fair die are yellow, two faces red and one blue. The die is tossed three times. The probability that t...
IIT-JEE 1992
If the mean and the variance of binomial variate $$X$$ are $$2$$ and $$1$$ respectively, then the probability that $$X$$...
IIT-JEE 1991
Let $$A$$ and $$B$$ be two events such that $$P\,\,\left( A \right)\,\, = \,\,0.3$$ and $$P\left( {A \cup B} \right) = 0...
IIT-JEE 1990
A pair of fair dice is rolled together till a sum of either $$5$$ or $$7$$ is obtained. Then the probability that $$5$$ ...
IIT-JEE 1989
Urn $$A$$ contains $$6$$ red and $$4$$ black balls and urn $$B$$ contains $$4$$ red and $$6$$ black balls. One ball is d...
IIT-JEE 1988
If $${{1 + 3p} \over 3},\,\,\,{{1 - p} \over 4}$$ and $$\,{{1 - 2p} \over 2}$$ are the probabilities of three mutually e...
IIT-JEE 1986
$$P\left( {A \cup B} \right) = P\left( {A \cap B} \right)$$ if and only if the relation between $$P(A)$$ and $$P(B)$$ is...
IIT-JEE 1985
A box contains $$100$$ tickets numbered $$1, 2, ....., 100.$$ Two tickets are chosen at random. It is given that the max...
IIT-JEE 1985
For a biased die the probabilities for the different faces to turn up are given below : This die tossed and you are to...
IIT-JEE 1981

True or False

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

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