Probability
652 Questions
Start JEE Mains Test
1996
Q601
JEE Advanced
MCQ
14 Mar 2026
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 $C$ occurs)$=P$ (exactly one of the events $C$ or $A$ occurs)$=p$ and $P$ (all the three events occur simultaneously) $ = {p^2},$ where $0 < p < 1/2.$ Then the probability of at least one of the three events $A,B$ and $C$ occurring is
A.
${{3p + 2{p^2}} \over 2}$
B.
${{p + 3{p^2}} \over 4}$
C.
${{p + 3{p^2}} \over 2}$
D.
${{3p + 2{p^2}} \over 4}$
1996
Q602
JEE Advanced
Numerical
14 Mar 2026
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 seating arrangements are possible if $3$ girls should sit together in a back row on adjacent seats? Now, if all the seating arrangements are equally likely, what is the probability of $3$ girls sitting together in a back row on adjacent seats?
Correct Answer: $$7\left( {13!} \right),12!,1/9!$$
1995
Q603
JEE Advanced
MCQ
14 Mar 2026
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 $5$ match series India's second win occurs at third test is
A.
$1/8$
B.
$1/4$
C.
$1/2$
D.
$2/3$
1995
Q604
JEE Advanced
MCQ
14 Mar 2026
Three of six vertices of a regular hexagon are chosen at random. The probability that the triangle with three vertices is equilateral, equals
A.
$1/2$
B.
$1/5$
C.
$1/10$
D.
$1/20$
1995
Q605
JEE Advanced
MSQ
14 Mar 2026
Let $0 < P\left( A \right) < 1,0 < P\left( B \right) < 1$ and
$P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( A \right)P\left( B \right)$ then
$P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( A \right)P\left( B \right)$ then
A.
$P\left( {B/A} \right) = P\left( B \right) - P\left( A \right)$
B.
$P\left( {A' - B'} \right) = P\left( {A'} \right) - P\left( {B'} \right)$
C.
$P\left( {A \cup B} \right)' = P\left( {A'} \right) - P\left( {B'} \right)$
D.
$P\left( {A/B} \right) = P\left( A \right)$
1994
Q606
JEE Advanced
MCQ
14 Mar 2026
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 independent
${S_2}\,:\,A$ and $B \cap C$ are independent
Then,
${S_1}\,:\,A$ and $B \cup C$ are independent
${S_2}\,:\,A$ and $B \cap C$ are independent
Then,
A.
Both ${S_1}$ and ${S_2}$ are true
B.
Only ${S_1}$ is true
C.
Only ${S_2}$ is true
D.
Neither ${S_1}$ nor ${S_2}$ is true
1994
Q607
JEE Advanced
Numerical
14 Mar 2026
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 is noted. If the result is a tail, a card from a well shuffled pack of eleven cards numbered $2, 3,4,.....12$ is picked and the number on the card is noted. What is the probability that the noted number is either $7$ or $8$?
Correct Answer: $$0.2436$$
1994
Q608
JEE Advanced
Numerical
14 Mar 2026
If two events $A$ and $B$ are such that $P\,\,\left( {{A^c}} \right)\,\, = \,\,0.3,\,\,P\left( B \right) = 0.4$ and $P\left( {A \cap {B^c}} \right) = 0.5,$ then $P\left( {B/\left( {A \cup {B^c}} \right)} \right.$$\left. \, \right] = $ ............
Correct Answer: $$1/4$$
1993
Q609
JEE Advanced
MCQ
14 Mar 2026
An unbiased die with faces marked $1,2,3,4,5$ and $6$ is rolled four times. Out of four face values obtained, the probability that the minimum face value is not less than $2$ and the maximum face value is not greater than $5,$ is then:
A.
$16/81$
B.
$1/81$
C.
$80/81$
D.
$65/81$
1993
Q610
JEE Advanced
MSQ
14 Mar 2026
$E$ and $F$ are two independent events. The probability that both $E$ and $F$ happen is $1/12$ and the probability that neither $E$ nor $F$ happens is $1/2.$ Then,
A.
$\,P\left( E \right) = 1/3,P\left( F \right) = 1/4$
B.
$\,P\left( E \right) = 1/2,P\left( F \right) = 1/6$
C.
$\,P\left( E \right) = 1/6,P\left( F \right) = 1/2$
D.
$\,P\left( E \right) = 1/4,P\left( F \right) = 1/3$
1993
Q611
JEE Advanced
Numerical
14 Mar 2026
Numbers are selected at random, one at a time, from the two- digit numbers $00, 01, 02 ......, 99$ with replacement. An event $E$ occurs if only if the product of the two digits of a selected number is $18$. If four numbers are selected, find probability that the event $E$ occurs at least $3$ times.
Correct Answer: $${{97} \over {{{\left( {25} \right)}^4}}}$$
1992
Q612
JEE Advanced
MCQ
14 Mar 2026
India plays two matches each with West Indies and Australia. In any match the probabilities of India getting, points $0,$ $1$ and $2$ are $0.45, 0.05$ and $0.50$ respectively. Assuming that the outcomes are independent, the probability of India getting at least $7$ points is
A.
$0.8750$
B.
$0.0875$
C.
$0.0625$
D.
$0.0250$
1992
Q613
JEE Advanced
Numerical
14 Mar 2026
A lot contains $50$ defective and $50$ non defective bulbs. Two bulbs are drawn at random, one at a time, with replacement. The events $A, B, C$ are defined as
$A=$ (the first bulbs is defective)
$B=$ (the second bulbs is non-defective)
$C=$ (the two bulbs are both defective or both non defective)
Determine whether
(i) $\,\,\,\,\,$ $A, B, C$ are pairwise independent
(ii)$\,\,\,\,\,$ $A, B, C$ are independent
$A=$ (the first bulbs is defective)
$B=$ (the second bulbs is non-defective)
$C=$ (the two bulbs are both defective or both non defective)
Determine whether
(i) $\,\,\,\,\,$ $A, B, C$ are pairwise independent
(ii)$\,\,\,\,\,$ $A, B, C$ are independent
Correct Answer: $$A, B, C$$ are pairwise independent but $$A, B, C$$ are dependent.
1992
Q614
JEE Advanced
Numerical
14 Mar 2026
Three faces of a fair die are yellow, two faces red and one blue. The die is tossed three times. The probability that the colours, yellow, red and blue, appear in the first, second and the third tosses respectively is ...............
Correct Answer: $$1/36$$
1991
Q615
JEE Advanced
MSQ
14 Mar 2026
For any two events $A$ and $B$ in a simple space
A.
$P\left( {A/B} \right) \ge {{P\left( A \right) + P\left( B \right) - 1} \over {P\left( B \right)}},P\left( B \right) \ne 0$ is always true
B.
$P\left( {A \cap \overline B } \right) = P\left( A \right) - P\left( {A \cap B} \right)\,\,$ does not hold
C.
$P\left( {A \cup B} \right) = 1 - P\left( {\overline A } \right)P\left( {\overline B } \right),$ if $A$ and $B$ are independent
D.
$P\left( {A \cup B} \right) = 1 - P\left( {\overline A } \right)P\left( {\overline B } \right),$ if $A$ and $B$ are disjoint.
1991
Q616
JEE Advanced
Numerical
14 Mar 2026
In a test an examine either guesses or copies or knows the answer to a multiple choice question with four choices. The probability that he make a guess is $1/3$ and the probability that he copies the answer is $1/6$. The probability that his answer is correct given that he copied it, is $1/8$. Find the probability that he knew the answer to the questions given that he correctly answered it.
Correct Answer: $$24/29$$
1991
Q617
JEE Advanced
Numerical
14 Mar 2026
If the mean and the variance of binomial variate $X$ are $2$ and $1$ respectively, then the probability that $X$ takes a value greater than one is equal to ...............
Correct Answer: $$11/16$$
1990
Q618
JEE Advanced
Numerical
14 Mar 2026
A is a set containing $n$ elements. $A$ subset $P$ of $A$ is chosen at random. The set $A$ is reconstructed by replacing the elements of $P.$ $A$ subset $Q$ of $A$ is again chosen at random. Find the probability that $P$ and $Q$ have no common elements.
Correct Answer: $${\left( {{3 \over 4}} \right)^n}$$
1990
Q619
JEE Advanced
Numerical
14 Mar 2026
Let $A$ and $B$ be two events such that $P\,\,\left( A \right)\,\, = \,\,0.3$ and $P\left( {A \cup B} \right) = 0.8.$ If $A$ and $B$ are independent events then $P(B)=$ ................
Correct Answer: $$5/7$$
1989
Q620
JEE Advanced
MSQ
14 Mar 2026
If $E$ and $F$ are independent events such that $0 < P\left( E \right) < 1$ and $0 < P\left( F \right) < 1,$ then
A.
$E$ and $F$ are mutually exclusive
B.
$E$ and ${F^c}$ (the complement of the event $F$) are independent
C.
${E^c}$ and ${F^c}$ are independent
D.
$P\left( {E|F} \right) + P\left( {{E^c}|F} \right) = 1.$
1989
Q621
JEE Advanced
Numerical
14 Mar 2026
Suppose the probability for A to win a game against B is $0.4.$ If $A$ has an option of playing either a "best of $3$ games" or a "best of $5$ games" match against $B$, which option should be choose so that the probability of his winning the match is higher ? (No game ends in a draw).
Correct Answer: best of $$3$$ games
1989
Q622
JEE Advanced
Numerical
14 Mar 2026
A pair of fair dice is rolled together till a sum of either $5$ or $7$ is obtained. Then the probability that $5$ comes before $7$ is ...............
Correct Answer: $$2/5$$
1989
Q623
JEE Advanced
MCQ
14 Mar 2026
If the probability for $A$ to fail in an examination is $0.2$ and that for $B$ is $0.3$, then the probability that either $A$ or $B$ fails is $0.5$
A.
TRUE
B.
FALSE
1988
Q624
JEE Advanced
MCQ
14 Mar 2026
One hundred identical coins, each with probability, $p,$ of showing up heads are tossed once. If $0 < p < 1$ and the probability of heads showing on $50$ coins is equal to that of heads showing on $51$ coins, then the value of $p$ is
A.
$1/2$
B.
$49/101$
C.
$50/101$
D.
$51/101.$
1988
Q625
JEE Advanced
MSQ
14 Mar 2026
For two given events $A$ and $B,$ $P\left( {A \cap B} \right)$
A.
not less than $P\left( A \right) + P\left( B \right) - 1$
B.
not greater than $P\left( A \right) + P\left( B \right)$
C.
equal to $P\left( A \right) + P\left( B \right) - P\left( {A \cup B} \right)\,\,$
D.
$P\left( A \right) + P\left( B \right) + P\left( {A \cup B} \right)\,\,$
1988
Q626
JEE Advanced
Numerical
14 Mar 2026
A box contains $2$ fifty paise coins, $5$ twenty five paise coins and a certain fixed number $N\,\,\left( { \ge 2} \right)$ of ten and five paise coins. Five coins are taken out of the box at random. Find the probability that the total value of these $5$ coins is less than one rupee and fifty paise.
Correct Answer: $$\,1 - {{10\left( {N + 2} \right)} \over {N + 7{c_5}}}$$
1988
Q627
JEE Advanced
Numerical
14 Mar 2026
Urn $A$ contains $6$ red and $4$ black balls and urn $B$ contains $4$ red and $6$ black balls. One ball is drawn at random from urn $A$ and placed in urn $B$. The one ball is drawn at random from urn $B$ and placed in urn $A$. If one ball is now drawn at random from urn $A$, the probability that it is found to be red is ................
Correct Answer: $$32/55$$
1987
Q628
JEE Advanced
Numerical
14 Mar 2026
A man takes a step forward with probability $0.4$ and backwards with probability $0.6$ Find the probability that at the end of eleven steps he is one step away from the starting point.
Correct Answer: $$0.37$$
1986
Q629
JEE Advanced
MCQ
14 Mar 2026
A student appears for tests, $I$, $II$ and $III$. The student is successful if he passes either in tests $I$ and $II$ or tests $I$ and $III$. The probabilities of student passing in tests $I$, $II$ and $III$ are $p, q$ and ${1 \over 2}$ respectively. If the probability that the student is successful is ${1 \over 2}$, then
A.
$p=q=1$
B.
$p = q = {1 \over 2}$
C.
$p=1,$ $q=0$
D.
$p = 1,q = {1 \over 2}$
1986
Q630
JEE Advanced
MCQ
14 Mar 2026
The probability that at least one of the events $A$ and $B$ occurs is $0.6$. If $A$ and $B$ occur simultaneously with probability $0.2,$ then $P\left( {\overline A } \right) + P\left( {\overline B } \right)$ is
A.
$0.4$
B.
$0.8$
C.
$1.2$
D.
$1.4$
1986
Q631
JEE Advanced
Numerical
14 Mar 2026
A lot contains $20$ articles. The probability that the lot contains exactly $2$ defective articles is $0.4$ and the probability that the lot contains exactly $3$ defective articles is $0.6$. Articles are drawn from the lot at random one by one without replacement and are tested till all defective articles are found. What is the probability that the testing procedure ends at the twelth testing.
Correct Answer: $$99/1900$$
1986
Q632
JEE Advanced
Numerical
14 Mar 2026
If ${{1 + 3p} \over 3},\,\,\,{{1 - p} \over 4}$ and $\,{{1 - 2p} \over 2}$ are the probabilities of three mutually exclusive events, then the set of all values of $p$ is ..............
Correct Answer: $${1 \over 3} \le p \le {1 \over 2}$$
1985
Q633
JEE Advanced
Numerical
14 Mar 2026
In a multiple-choice question there are four alternative answers, of which one or more are correct. A candidate will get marks in the question only if he ticks the correct answers. The candidate decides to tick the answers at random, If he is allowed upto three chances to answer the questions, find the probability that he will get marks in the questions.
Correct Answer: $$1/5$$
1985
Q634
JEE Advanced
Numerical
14 Mar 2026
A box contains $100$ tickets numbered $1, 2, ....., 100.$ Two tickets are chosen at random. It is given that the maximum number on the two chosen tickets is not more than $10.$ The minimum number on them is $5$ with probability ........
Correct Answer: $$1/9$$
1985
Q635
JEE Advanced
Numerical
14 Mar 2026
$P\left( {A \cup B} \right) = P\left( {A \cap B} \right)$ if and only if the relation between $P(A)$ and $P(B)$ is .............
Correct Answer: $$P(A)$$ $$=$$ $$P(B)$$
1984
Q636
JEE Advanced
MCQ
14 Mar 2026
A box contains $24$ identical balls of which $12$ are white and $12$ are black. The balls are drawn at random from the box one at a time with replacement. The probability that a white ball is drawn for the $4$th time on the $7$th draw is
A.
$5/64$
B.
$27/32$
C.
$5/32$
D.
$1/2$
1984
Q637
JEE Advanced
MCQ
14 Mar 2026
Three identical dice are rolled. The probability that the same number will appear on each of them is
A.
$1/6$
B.
$1/36$
C.
$1/18$
D.
$3/28$
1984
Q638
JEE Advanced
MSQ
14 Mar 2026
If $M$ and $N$ are any two events, the probability that exactly one of them occurs is
A.
$P\left( M \right) + P\left( N \right) - 2P\left( {M \cap N} \right)$
B.
$P\left( M \right) + P\left( N \right) - P\left( {M \cap N} \right)$
C.
$P\left( {{M^c}} \right) + P\left( {{N^c}} \right) - 2P\left( {{M^c} \cap {N^c}} \right)$
D.
$P\left( {M \cap {N^c}} \right) + P\left( {{M^c} \cap N} \right)$
1984
Q639
JEE Advanced
Numerical
14 Mar 2026
In a certain city only two newspapers $A$ and $B$ are published, it is known that $25$% of the city population reads $A$ and $20$% reads $B$ while $8$% reads both $A$ and $B$. It is also known that $30$% of those who read $A$ but not $B$ look into advertisements and $40$% of those who read $B$ but not $A$ look into advertisements while $50$% of those who read both $A$ and $B$ look into advertisements. What is the percentage of the population that reads an advertisement?
Correct Answer: $$13.9$$%
1983
Q640
JEE Advanced
MCQ
14 Mar 2026
Fifteen coupons are numbered $1, 2 ........15,$ respectively. Seven coupons are selected at random one at a time with replacement. The probability that the largest number appearing on a selected coupon is $9,$ is
A.
${\left( {{9 \over {16}}} \right)^6}$
B.
${\left( {{18 \over {15}}} \right)^7}$
C.
${\left( {{3 \over {5}}} \right)^7}$
D.
none of these
1983
Q641
JEE Advanced
Numerical
14 Mar 2026
$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} \right) = 0.28;\,\,P\left( {ABC} \right) = 0.09$
$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} \right) = 0.28;\,\,P\left( {ABC} \right) = 0.09$
If $P\left( {A \cup B \cup C} \right) \ge 0.75,$ then show that $P$ $(BC)$ lies in the interval $0.23 \le x \le 0.48$
Correct Answer: Solve it.
1983
Q642
JEE Advanced
Numerical
14 Mar 2026
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$ is the number of cards required to be drawn, then show that ${P_r}\left\{ {N = n} \right\} = {{\left( {n - 1} \right)\left( {52 - n} \right)\left( {51 - n} \right)} \over {50 \times 49 \times 17 \times 13}}$ where $2 \le n \le 50$
Correct Answer: Solve it.
1983
Q643
JEE Advanced
MCQ
14 Mar 2026
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$
A.
TRUE
B.
FALSE
1982
Q644
JEE Advanced
MCQ
14 Mar 2026
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 {\overline B }}} \right)$ is equal to
A.
$1 - P({A \over B})$ (Here $\overline A $ and $\overline B $ are complements of $A$ and $B$ respectively).
B.
$1 - P({{\overline A } \over B})$ (Here $\overline A $ and $\overline B $ are complements of $A$ and $B$ respectively).
C.
${{1 - P\left( {A \cup B} \right)} \over {P\left( {\overline B } \right)}}$ (Here $\overline A $ and $\overline B $ are complements of $A$ and $B$ respectively).
D.
${{P\left( {\overline A } \right)} \over {P\left( {\overline B } \right)}}$ (Here $\overline A $ and $\overline B $ are complements of $A$ and $B$ respectively).
1982
Q645
JEE Advanced
Numerical
14 Mar 2026
$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$ and $B$ are selected is atmost $0.3$. Is it possible that the probability of $B$ getting selected is $0.9$ ?
Correct Answer: No.
1981
Q646
JEE Advanced
Numerical
14 Mar 2026
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, second, third and fourth shot are $0.4, 0.3, 0.2$ and $0.1$ respectively. What is the probability that the gun hits the plane?
Correct Answer: $$0.69$$
1981
Q647
JEE Advanced
Numerical
14 Mar 2026
For a biased die the probabilities for the different faces to turn up are given below :
This die tossed and you are told that either face $1$ or face $2$ has turned up. Then the probability that it is face $1$ is ...............
Correct Answer: $${5 \over {21}}$$
1980
Q648
JEE Advanced
MCQ
14 Mar 2026
Two events $A$ and $B$ have probabilities $0.25$ and $0.50$ respectively. The probability that both $A$ and $B$ occur simultaneously is $0.14$. Then the probability that neither $A$ nor $B$ occurs is
A.
$0.39$
B.
$0.25$
C.
$0.11$
D.
none of these
1980
Q649
JEE Advanced
MCQ
14 Mar 2026
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 probability that the event $A$ happens at least once is
A.
$0.936$
B.
$0.784$
C.
$0.904$
D.
none of these
1979
Q650
JEE Advanced
MCQ
14 Mar 2026
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 number. The two events $x$ and $y$ are:
A.
Mutually exclusive
B.
Independent and mutually exclusive
C.
Dependent
D.
None of these.