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.
Correct Answer: $$2{p^2} - {p^3}$$
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?
Correct Answer: $${{9m} \over {m + 8N}}$$
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?
Correct Answer: $${{6{c_3}\left[ {{3^n} - 3\left( {{2^n}} \right) + 3} \right]} \over {{6^n}}}$$
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?
Correct Answer: $$\,{m \over {m + n}}$$
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 $n \ge 3.$
Correct Answer: Solve it.
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?
Correct Answer: $$4/35$$
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 $p$).
Correct Answer: $$\alpha = {p \over {1 - {{\left( {1 - p} \right)}^3}}},$$ $$\beta = {{\left( {1 - p} \right)p} \over {1 - {{\left( {1 - p} \right)}^3}}},$$ $$\gamma = {{p{{\left( {1 - p} \right)}^2}} \over {1 - {{\left( {1 - p} \right)}^3}}}\,\,$$
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$f(x).$
Correct Answer: $$f\left( x \right) = {x^3} - {x^2}$$
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.
Correct Answer: $$0.62$$
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?
Correct Answer: $$7\left( {13!} \right),12!,1/9!$$
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 $8$?
Correct Answer: $$0.2436$$
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.
Correct Answer: $${{97} \over {{{\left( {25} \right)}^4}}}$$
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
$A=$ (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.
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.
Correct Answer: $$24/29$$
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.
Correct Answer: $${\left( {{3 \over 4}} \right)^n}$$
1989
JEE Advanced
Numerical
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 $5$ 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
1988
JEE Advanced
Numerical
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 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: $$\,1 - {{10\left( {N + 2} \right)} \over {N + 7{c_5}}}$$
1987
JEE Advanced
Numerical
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 one step away from the starting point.
Correct Answer: $$0.37$$
1986
JEE Advanced
Numerical
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 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: $$99/1900$$
1985
JEE Advanced
Numerical
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 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: $$1/5$$
1984
JEE Advanced
Numerical
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 $B$ 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
JEE Advanced
Numerical
IIT-JEE 1983
$A, B, C$ are events such that
$P\left( A \right) = 0.3,P\left( B \right) = 0.4,P\left( C \right) = 0.8$
$P\left( {AB} \right) = 0.08,P\left( {AC} \right) = 0.28;\,\,P\left( {ABC} \right) = 0.09$
$P\left( A \right) = 0.3,P\left( B \right) = 0.4,P\left( C \right) = 0.8$
$P\left( {AB} \right) = 0.08,P\left( {AC} \right) = 0.28;\,\,P\left( {ABC} \right) = 0.09$
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
JEE Advanced
Numerical
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 $N$ 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.
1982
JEE Advanced
Numerical
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 $A$ 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
JEE Advanced
Numerical
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, 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: $$0.69$$
1979
JEE Advanced
Numerical
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.
(i) the six girls sit together
(ii) the boys and girls sit alternately.
Correct Answer: (i) $${1 \over {132}}$$
<br>(ii) $${1 \over {462}}$$
1978
JEE Advanced
Numerical
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 the probability that the balls drawn are in the order $2$ black, $4$ white and $3$ red.
Correct Answer: $${1 \over {1260}}$$
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.$$\left. \, \right] = $ ............
Correct Answer: $$1/4$$
1992
JEE Advanced
Numerical
IIT-JEE 1992
Three faces of a fair die are yellow, two faces red and one blue. The die is tossed three times. The probability that the colours, yellow, red and blue, appear in the first, second and the third tosses respectively is ...............
Correct Answer: $$1/36$$
1991
JEE Advanced
Numerical
IIT-JEE 1991
If the mean and the variance of binomial variate $X$ are $2$ and $1$ respectively, then the probability that $X$ takes a value greater than one is equal to ...............
Correct Answer: $$11/16$$
1990
JEE Advanced
Numerical
IIT-JEE 1990
Let $A$ and $B$ be two events such that $P\,\,\left( A \right)\,\, = \,\,0.3$ and $P\left( {A \cup B} \right) = 0.8.$ If $A$ and $B$ are independent events then $P(B)=$ ................
Correct Answer: $$5/7$$
1989
JEE Advanced
Numerical
IIT-JEE 1989
A pair of fair dice is rolled together till a sum of either $5$ or $7$ is obtained. Then the probability that $5$ comes before $7$ is ...............
Correct Answer: $$2/5$$
1988
JEE Advanced
Numerical
IIT-JEE 1988
Urn $A$ contains $6$ red and $4$ black balls and urn $B$ contains $4$ red and $6$ black balls. One ball is drawn at random from urn $A$ and placed in urn $B$. The one ball is drawn at random from urn $B$ and placed in urn $A$. If one ball is now drawn at random from urn $A$, the probability that it is found to be red is ................
Correct Answer: $$32/55$$
1986
JEE Advanced
Numerical
IIT-JEE 1986
If ${{1 + 3p} \over 3},\,\,\,{{1 - p} \over 4}$ and $\,{{1 - 2p} \over 2}$ are the probabilities of three mutually exclusive events, then the set of all values of $p$ is ..............
Correct Answer: $${1 \over 3} \le p \le {1 \over 2}$$
1985
JEE Advanced
Numerical
IIT-JEE 1985
A box contains $100$ tickets numbered $1, 2, ....., 100.$ Two tickets are chosen at random. It is given that the maximum number on the two chosen tickets is not more than $10.$ The minimum number on them is $5$ with probability ........
Correct Answer: $$1/9$$
1985
JEE Advanced
Numerical
IIT-JEE 1985
$P\left( {A \cup B} \right) = P\left( {A \cap B} \right)$ if and only if the relation between $P(A)$ and $P(B)$ is .............
Correct Answer: $$P(A)$$ $$=$$ $$P(B)$$
1981
JEE Advanced
Numerical
IIT-JEE 1981
For a biased die the probabilities for the different faces to turn up are given below :
This die tossed and you are told that either face $1$ or face $2$ has turned up. Then the probability that it is face $1$ is ...............
Correct Answer: $${5 \over {21}}$$
1989
JEE Advanced
MCQ
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 is $0.5$
A.
TRUE
B.
FALSE
1983
JEE Advanced
MCQ
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 $1/35$
A.
TRUE
B.
FALSE