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## Probability

### Mathematics

#### Previous Years Questions

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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...
JEE Advanced 2022 Paper 2 Online
Two players, $$P_{1}$$ and $$P_{2}$$, play a game against each other. In every round of the game, each player rolls a fa...
JEE Advanced 2022 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 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}$$being white is IIT-JEE 2011 Paper 1 Offline A signal which can be green or red with probability$${4 \over 5}$$and$${1 \over 5}$$respectively, is received by sta... IIT-JEE 2010 Paper 2 Offline Let$$\omega $$be a complex cube root of unity with$$\omega \ne 1.$$A fair die is thrown three times. If$${r_1},$$... IIT-JEE 2010 Paper 1 Offline A fair die is tossed repeatedly until a six is obtained. Let$$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$$denote the number of tosses required. The conditiona... IIT-JEE 2009 A fair die is tossed repeatedly until a six is obtained. Let$$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) &gt; 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
If $$E$$ and $$F$$ are events with $$P\left( E \right) \le P\left( F \right)$$ and $$P\left( {E \cap F} \right) &gt; 0,... IIT-JEE 1998 There are four machines and it is known that exactly two of them are faulty. They are tested, one by one, in a random or... IIT-JEE 1998 If from each of the three boxes containing$$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
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
The probability of India winning a test match against West Indies is $$1/2$$. Assuming independence from match to match...
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 $$0 &lt; p &lt; 1$$ 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
Three identical dice are rolled. The probability that the same number will appear on each of them is
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...
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 $$P\left( A \right) &gt; 0,$$ 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

## Numerical

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In a study about a pandemic, data of 900 persons was collected. It was found that 190 persons had symptom of fever, 22...
JEE Advanced 2022 Paper 1 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 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 $$\alpha ,... JEE Advanced 2013 Paper 1 Offline ## MCQ (More than One Correct Answer) More Let E, F and G be three events having probabilities$$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 &lt; P\left( A \right) &lt; 1,0 &lt; P\left( B \right) &lt; 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 $$0 &lt; P\left( E \right) &lt; 1$$ and $$0 &lt; P\left( F \right) &... IIT-JEE 1989 For two given events$$A$$and$$B,P\left( {A \cap B} \right)$$IIT-JEE 1988 If$$M$$and$$N$$are any two events, the probability that exactly one of them occurs is IIT-JEE 1984 ## Subjective More A person goes to office either by car, scooter, bus or train, the probability of which being$${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
$$A, B, C$$ are events such that $$P\left( A \right) = 0.3,P\left( B \right) = 0.4,P\left( C \right) = 0.8$$ $$P\left( {... IIT-JEE 1983 Cards are drawn one by one at random from a well - shuffled full pack of$$52$$playing cards until$$2$$aces are obtai... IIT-JEE 1983$$A$$and$$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$$red balls till a... IIT-JEE 1978 ## Fill in the Blanks More If two events$$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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