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Probability

633 Questions
All Exams JEE Mains JEE Advanced TS-EAMCET AP-EAPCET
2004 JEE Mains MCQ
AIEEE 2004
The probability that $A$ speaks truth is ${4 \over 5},$ while the probability for $B$ is ${3 \over 4}.$ The probability that they contradict each other when asked to speak on a fact is :
A.
${4 \over 5}$
B.
${1 \over 5}$
C.
${7 \over 20}$
D.
${3 \over 20}$
2004 JEE Mains MCQ
AIEEE 2004
The mean and the variance of a binomial distribution are $4$ and $2$ respectively. Then the probability of $2$ successes is :
A.
${28 \over 256}$
B.
${219 \over 256}$
C.
${128 \over 256}$
D.
${37 \over 256}$
2004 JEE Advanced MCQ
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 both $2$ and $3$ is
A.
$4/25$
B.
$4/35$
C.
$4/33$
D.
$4/1155$
2004 JEE Advanced Numerical
IIT-JEE 2004
$A$ and $B$ are two independent events. $C$ is even in which exactly one of $A$ or $B$ occurs. Prove that $P\left( C \right) \ge P\left( {A \cup B} \right)P\left( {\overline A \cap \overline B } \right)$
2004 JEE Advanced Numerical
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 $4$ white balls, find the probability that exactly one white is drawn in the next two draws. (binomial coefficients can be left as such)
2003 JEE Mains MCQ
AIEEE 2003
Five horses are in a race. Mr. A selects two of the horses at random and bets on them. The probability that Mr. A selected the winning horse is :
A.
${{2 \over 5}}$
B.
${{4 \over 5}}$
C.
${{3 \over 5}}$
D.
${{1 \over 5}}$
2003 JEE Mains MCQ
AIEEE 2003
The mean and variance of a random variable $X$ having binomial distribution are $4$ and $2$ respectively, then $P(X=1)$ is :
A.
${1 \over 4}$
B.
${1 \over 32}$
C.
${1 \over 16}$
D.
${1 \over 8}$
2003 JEE Mains MCQ
AIEEE 2003
Events $A, B, C$ are mutually exclusive events such that $P\left( A \right) = {{3x + 1} \over 3},$ $P\left( B \right) = {{1 - x} \over 4}$ and $P\left( C \right) = {{1 - 2x} \over 2}$ The set of possible values of $x$ are in the interval.
A.
$\left[ {0,1} \right]$
B.
$\left[ {{1 \over 3},{1 \over 2}} \right]$
C.
$\left[ {{1 \over 3},{2 \over 3}} \right]$
D.
$\left[ {{1 \
3},{13 \over 3}} \right]$
2003 JEE Advanced MCQ
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 the two numbers is less than $4$ is
A.
$1/15$
B.
$14/15$
C.
$1/5$
D.
$4/5$
2003 JEE Advanced MCQ
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
$P\left( {\overline A \cap B \cap \overline C } \right) = {1 \over 3},\,\,$ then $P\left( {B \cap C} \right)$ is
A.
$1/12$
B.
$1/6$
C.
$1/15$
D.
$1/9$
2003 JEE Advanced Numerical
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 ${2 \over 3},{1 \over 2}$ and ${1 \over 3}$ respectively. If $A$ is hit then find the probability that $B$ hits the target and $C$ does not.
2003 JEE Advanced Numerical
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 the exams then the probability of his passing in the next exam is ${p \over 2}$ otherwise it remains the same. Find the probability that he will qualify.
2002 JEE Mains MCQ
AIEEE 2002
A problem in mathematics is given to three students $A,B,C$ and their respective probability of solving the problem is ${1 \over 2},{1 \over 3}$ and ${1 \over 4}.$ Probability that the problem is solved is :
A.
${3 \over 4}$
B.
${1 \over 2}$
C.
${2 \over 3}$
D.
${1 \over 3}$
2002 JEE Mains MCQ
AIEEE 2002
A dice is tossed $5$ times. Getting an odd number is considered a success. Then the variance of distribution of success is :
A.
$8/3$
B.
$3/8$
C.
$4/5$
D.
$5/4$
2002 JEE Mains MCQ
AIEEE 2002
$A$ and $B$ are events such that $P\left( {A \cup B} \right) = 3/4$,$P\left( {A \cap B} \right) = 1/4,$
$P\left( {\overline A } \right) = 2/3$ then $P\left( {\overline A \cap B} \right)$ is :
A.
$5/12$
B.
$3/8$
C.
$5/8$
D.
$1/4$
2002 JEE Advanced Numerical
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 $1/2$, while it is $2/3$ when a biased coin is tossed. A coin is drawn from the box at random and is tossed twice. The first time it shows head and the second time it shows tail. what is the probability that the coin drawn is fair?
2001 JEE Advanced Numerical
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 that, among the numbers $1,2,3,4,5,6,$ only three numbers appear in this list?
2001 JEE Advanced Numerical
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 same colour as that of the ball drawn. A ball is again drawn at random. What is the probability that the ball drawn now is white?
2000 JEE Advanced Numerical
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) consecutive heads occur. Prove that ${p_1} = 1,{p_2} = 1 - {p^2}$ and ${p_n} = \left( {1 - p} \right).\,\,{p_{n - 1}} + p\left( {1 - p} \right){p_{n - 2}}$ for all $n \ge 3.$
1999 JEE Advanced MCQ
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 divisible by $5$ equals
A.
$1/4$
B.
$1/7$
C.
$1/8$
D.
$1/49$
1999 JEE Advanced MSQ
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 a $75%$ chance of passing in at least one, a $50$% chance of passing in at least two, and a $40$% chance of passing in exactly two. Which of the following relations are true?
A.
$p+m+c=19/20$
B.
$p+m+c=27/20$
C.
$pmc=1/10$
D.
$pmc=1/4$
1999 JEE Advanced Numerical
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 ${P_i}$ will win if $i < j.$ Assuming that the players are paired at random in each round, what is the probability that the player ${P_4}$ reaches the final?
1998 JEE Advanced MCQ
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
A.
$1/2$
B.
$7/15$
C.
$2/15$
D.
$1/3$
1998 JEE Advanced MCQ
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 machines are identified. Then the probability that only two tests are needed is
A.
$1/3$
B.
$1/6$
C.
$1/2$
D.
$1/4$
1998 JEE Advanced MCQ
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
A.
$1/2$
B.
$1/32$
C.
$31/32$
D.
$1/5$
1998 JEE Advanced MCQ
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
A.
occurrence of $E$ $ \Rightarrow $ occurrence of $F$
B.
occurrence of $F$ $ \Rightarrow $ occurrence of $E$
C.
non-occurrence of $E$ $ \Rightarrow $ non-occurrence of $F$
D.
none of the above implications holds
1998 JEE Advanced MCQ
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 drawn at random, then the probability that $2$ white and $1$ black ball will be drawn is
A.
$13/32$
B.
$1/4$
C.
$1/32$
D.
$3/16$
1998 JEE Advanced MSQ
IIT-JEE 1998
If $\overline E $ and $\overline F $ are the complementary events of events $E$ and $F$ respectively and if $0 < P\left( F \right) < 1,$ then
A.
$P\left( {E/F} \right) + P\left( {\overline E /F} \right) = 1$
B.
$P\left( {E/F} \right) + P\left( {E/\overline F } \right) = 1$
C.
$P\left( {\overline E /F} \right) + P\left( {E/\overline F } \right) = 1$
D.
$P\left( {E/\overline F } \right) + P\left( {\overline E /\overline F } \right) = 1$
1998 JEE Advanced Numerical
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 probability that the coin shows a head. Let $\alpha ,\,\,\,\beta $ and $\gamma $ be, respectively, the probabilities that $A, B$ and $C$ gets the first head. Prove that $\beta = \left( {1 - p} \right)\alpha $ Determine $\alpha ,\beta $ and $\gamma $ (in terms of $p$).
1998 JEE Advanced Numerical
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 a function $y=f(x),$ $0 \le x \le 1,$ $f(0)=0.$ For a point $P$ on ${C_1},$ let the lines through $P,$ parallel to the axes, meet ${C_2}$ and ${C_3}$ at $Q$ and $R$ respectively (see figure.) If for every position of $P$ (on ${C_1}$ ), the areas of the shaded regions $OPQ$ and $ORP$ are equal, determine the function$f(x).$ IIT-JEE 1998 Mathematics - Probability Question 37 English
1997 JEE Advanced Numerical
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 roots of the equation ${x^2} + px + q = 0$ are real.
1996 JEE Advanced MCQ
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 $C$ 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 JEE Advanced Numerical
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 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?
1995 JEE Advanced MCQ
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 $5$ match series India's second win occurs at third test is
A.
$1/8$
B.
$1/4$
C.
$1/2$
D.
$2/3$
1995 JEE Advanced MCQ
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
A.
$1/2$
B.
$1/5$
C.
$1/10$
D.
$1/20$
1995 JEE Advanced MSQ
IIT-JEE 1995 Screening
Let $0 < P\left( A \right) < 1,0 < P\left( B \right) < 1$ and
$P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( A \right)P\left( B \right)$ 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 JEE Advanced MCQ
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 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 JEE Advanced Numerical
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 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 $8$?
1994 JEE Advanced Numerical
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 $P\left( {A \cap {B^c}} \right) = 0.5,$ then $P\left( {B/\left( {A \cup {B^c}} \right)} \right.$$\left. \, \right] = $ ............
1993 JEE Advanced MCQ
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 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 JEE Advanced MSQ
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 $F$ 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 JEE Advanced Numerical
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 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.
1992 JEE Advanced MCQ
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 $0.45, 0.05$ 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 JEE Advanced Numerical
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 $A, B, C$ 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
1992 JEE Advanced Numerical
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 blue, appear in the first, second and the third tosses respectively is ...............
1991 JEE Advanced MSQ
IIT-JEE 1991
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 JEE Advanced Numerical
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 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.
1991 JEE Advanced Numerical
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 one is equal to ...............
1990 JEE Advanced Numerical
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 $P.$ $A$ subset $Q$ of $A$ is again chosen at random. Find the probability that $P$ and $Q$ have no common elements.
1990 JEE Advanced Numerical
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 independent events then $P(B)=$ ................