Question 1: Palette 1/84

JEE Advanced 2026 MSQ

Suppose that Box I contains 6 red balls and 9 green balls, and Box II contains 8 red balls and 12 green balls. All the balls of Box I and Box II are mixed together and a ball is chosen at random from them. Let $E_1$ be the event that the ball chosen belonged to Box I and let $E_2$ be the event that the ball chosen belonged to Box II. Let $F_1$ be the event that the ball chosen is red and let $F_2$ be the event that the ball chosen is green.

Then which of the following statements is (are) TRUE?

A

The events $E_1$ and $F_1$ are independent

B

The events $E_2$ and $F_2$ are dependent

C

The conditional probability $P(F_1|E_1)$ is equal to the conditional probability $P(F_1|E_2)$

D

The conditional probability $P(F_1|E_1)$ is greater than the conditional probability $P(F_2|E_2)$

Question 2: Palette 2/84

JEE Advanced 2025 MCQ

Three students $S_1, S_2,$ and $S_3$ are given a problem to solve. Consider the following events:

U: At least one of $S_1, S_2,$ and $S_3$ can solve the problem,

V: $S_1$ can solve the problem, given that neither $S_2$ nor $S_3$ can solve the problem,

W: $S_2$ can solve the problem and $S_3$ cannot solve the problem,

T: $S_3$ can solve the problem.

For any event $E$, let $P(E)$ denote the probability of $E$. If

$P(U) = \dfrac{1}{2}$ , $P(V) = \dfrac{1}{10}$ , and $P(W) = \dfrac{1}{12}$,

then $P(T)$ is equal to

A

$\dfrac{13}{36}$

B

$\dfrac{1}{3}$

C

$\dfrac{19}{60}$

D

$\dfrac{1}{4}$

Question 3: Palette 3/84

JEE Advanced 2024 MCQ

A student appears for a quiz consisting of only true-false type questions and answers all the questions. The student knows the answers of some questions and guesses the answers for the remaining questions. Whenever the student knows the answer of a question, he gives the correct answer. Assume that the probability of the student giving the correct answer for a question, given that he has guessed it, is $\frac{1}{2}$. Also assume that the probability of the answer for a question being guessed, given that the student's answer is correct, is $\frac{1}{6}$. Then the probability that the student knows the answer of a randomly chosen question is :

A

$\frac{1}{12}$

B

$\frac{1}{7}$

C

$\frac{5}{7}$

D

$\frac{5}{12}$

Question 4: Palette 4/84

JEE Advanced 2023 MCQ

Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is $\frac{1}{3}$, then the probability that the experiment stops with head is :

A

$\frac{1}{3}$

B

$\frac{5}{21}$

C

$\frac{4}{21}$

D

$\frac{2}{7}$

Question 5: Palette 5/84

JEE Advanced 2023 MCQ

Let $X=\left\{(x, y) \in \mathbb{Z} \times \mathbb{Z}: \frac{x^2}{8}+\frac{y^2}{20}<1\right.$ and $\left.y^2<5 x\right\}$. Three distinct points $P, Q$ and $R$ are randomly chosen from $X$. Then the probability that $P, Q$ and $R$ form a triangle whose area is a positive integer, is :

A

$\frac{71}{220}$

B

$\frac{73}{220}$

C

$\frac{79}{220}$

D

$\frac{83}{220}$

Question 6: Palette 6/84

JEE Advanced 2022 MCQ

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 contains 1 blue, 12 green and 3 yellow balls,

Box-IV contains 10 green, 16 orange and 6 white balls.

A ball is chosen randomly from Box-I; call this ball $b$. If $b$ is red then a ball is chosen randomly from Box-II, if $b$ is blue then a ball is chosen randomly from Box-III, and if $b$ is green then a ball is chosen randomly from Box-IV. The conditional probability of the event 'one of the chosen balls is white' given that the event 'at least one of the chosen balls is green' has happened, is equal to

A

$\frac{15}{256}$

B

$\frac{3}{16}$

C

$\frac{5}{52}$

D

$\frac{1}{8}$

Question 7: Palette 7/84

JEE Advanced 2022 MCQ

Two players, $P_{1}$ and $P_{2}$, play a game against each other. In every round of the game, each player rolls a fair die once, where the six faces of the die have six distinct numbers. Let $x$ and $y$ denote the readings on the die rolled by $P_{1}$ and $P_{2}$, respectively. If $x>y$, then $P_{1}$ scores 5 points and $P_{2}$ scores 0 point. If $x=y$, then each player scores 2 points. If $x < y$, then $P_{1}$ scores 0 point and $P_{2}$ scores 5 points. Let $X_{i}$ and $Y_{i}$ be the total scores of $P_{1}$ and $P_{2}$, respectively, after playing the $i^{\text {th }}$ round.

List-I	List-II
(I) Probability of $\left(X_{2} \geq Y_{2}\right)$ is	(P) $\frac{3}{8}$
(II) Probability of $\left(X_{2}>Y_{2}\right)$ is	(Q) $\frac{11}{16}$
(III) Probability of $\left(X_{3}=Y_{3}\right)$ is	(R) $\frac{5}{16}$
(IV) Probability of $\left(X_{3}>Y_{3}\right)$ is	(S) $\frac{355}{864}$
	(T) $\frac{77}{432}$

The correct option is:

A

(I) $\rightarrow$ (Q); (II) $\rightarrow$ (R); (III) $\rightarrow$ (T); (IV) $\rightarrow(S)$

B

(I) $\rightarrow$ (Q); (II) $\rightarrow$ (R); (III) $\rightarrow$ (T); (IV) $\rightarrow$ (T)

C

(I) $\rightarrow$ (P); (II) $\rightarrow$ (R); (III) $\rightarrow(\mathrm{Q}) ;(\mathrm{IV}) \rightarrow(\mathrm{S})$

D

(I) $\rightarrow$ (P); (II) $\rightarrow$ (R); (III) $\rightarrow$ (Q); (IV) $\rightarrow$ (T)

Question 8: Palette 8/84

JEE Advanced 2021 MCQ

Consider three sets E₁ = {1, 2, 3}, F₁ = {1, 3, 4} and G₁ = {2, 3, 4, 5}. Two elements are chosen at random, without replacement, from the set E₁, and let S₁ denote the set of these chosen elements. Let E₂ = E₁ $-$ S₁ and F₂ = F₁ $\cup$ S₁. Now two elements are chosen at random, without replacement, from the set F₂ and let S₂ denote the set of these chosen elements.

Let G₂ = G₁ $\cup$ S₂. Finally, two elements are chosen at random, without replacement, from the set G₂ and let S₃ denote the set of these chosen elements.

Let E₃ = E₂ $\cup$ S₃. Given that E₁ = E₃, let p be the conditional probability of the event S₁ = {1, 2}. Then the value of p is

A

${1 \over 5}$

B

${3 \over 5}$

C

${1 \over 2}$

D

${2 \over 5}$

Question 9: Palette 9/84

JEE Advanced 2021 MSQ

Let E, F and G be three events having probabilities $P(E) = {1 \over 8}$, $P(F) = {1 \over 6}$ and $P(G) = {1 \over 4}$, and let P (E $\cap$ F $\cap$ G) = ${1 \over {10}}$. For any event H, if H^c denotes the complement, then which of the following statements is (are) TRUE?

A

$P(E \cap F \cap {G^c}) \le {1 \over {40}}$

B

$P({E^c} \cap F \cap G) \le {1 \over {15}}$

C

$P(E \cup F \cup G) \le {{13} \over {24}}$

D

$P({E^c} \cup {F^c} \cup {G^c}) \le {5 \over {12}}$

Question 10: Palette 10/84

JEE Advanced 2020 MCQ

Let C₁ and C₂ be two biased coins such that the probabilities of getting head in a single toss are ${{2 \over 3}}$ and ${{1 \over 3}}$, respectively. Suppose $\alpha $ is the number of heads that appear when C₁ is tossed twice, independently, and suppose $\beta $ is the number of heads that appear when C₂ is tossed twice, independently. Then the probability that the roots of the quadratic polynomial x² $-$ ax + $\beta $ are real and equal, is

A

${{40} \over {81}}$

B

${{20} \over {81}}$

C

${{1} \over {2}}$

D

${{1} \over {4}}$

Question 11: Palette 11/84

JEE Advanced 2019 MSQ

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, and B₃ contains 5 red and 3 green balls. Bags B₁, B₂ and B₃ have probabilities ${3 \over {10}}$, ${3 \over {10}}$ and ${4 \over {10}}$ respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?

A

Probability that the chosen ball is green, given that the selected bag is B₃, equals ${3 \over 8}$.

B

Probability that the selected bag is B₃, given that the chosen ball is green, equals ${5 \over 13}$.

C

Probability that the chosen ball is green equals ${39 \over 80}$.

D

Probability that the selected bag is B₃ and the chosen ball is green equals ${3 \over 10}$.

Question 12: Palette 12/84

JEE Advanced 2018 MCQ

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 arranged in a row, where initially the seat R_i is allotted to the student S_i, i = 1, 2, 3, 4, 5. But, on the examination day, the five students are randomly allotted the five seats.

(There are two questions based on Paragraph "A", the question given below is one of them)

The probability that, on the examination day, the student S₁ gets the previously allotted seat R₁, and NONE of the remaining students gets the seat previously allotted to him/her is

A

${3 \over {40}}$

B

${1 \over 8}$

C

${7 \over 40}$

D

${1 \over 5}$

Question 13: Palette 13/84

JEE Advanced 2018 MCQ

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 arranged in a row, where initially the seat R_i is allotted to the student S_i, i = 1, 2, 3, 4, 5. But, on the examination day, the five students are randomly allotted the five seats.

(There are two questions based on Paragraph "A", the question given below is one of them)

For i = 1, 2, 3, 4, let T_i denote the event that the students S_i and S_i+1 do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event ${T_1} \cap {T_2} \cap {T_3} \cap {T_4}$ is

A

${1 \over {15}}$

B

${1 \over {10}}$

C

${7 \over {60}}$

D

${1 \over {5}}$

Question 14: Palette 14/84

JEE Advanced 2017 MCQ

Three randomly chosen nonnegative integers x, y and z are found to satisfy the equation x + y + z = 10. Then the probability that z is even, is

A

${1 \over {2}}$

B

${36 \over {55}}$

C

${6 \over {11}}$

D

${5 \over {11}}$

Question 15: Palette 15/84

JEE Advanced 2017 MSQ

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

A

$P(Y) = {4 \over {15}}$

B

$P(X'|Y) = {1 \over 2}$

C

$P(X \cup Y) = {2 \over 5}$

D

$P(X \cap Y) = {1 \over 5}$

Question 16: Palette 16/84

JEE Advanced 2016 MCQ

Football teams ${T_1}$ and ${T_2}$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of ${T_1}$ winning, drawing and losing a game against ${T_2}$ are ${1 \over 2},{1 \over 6}$ and ${1 \over 3}$ respectively. Each team gets $3$ points for a win, $1$ point for a draw and $0$ point for a loss in a game. Let $X$ and $Y$ denote the total points scored by teams ${T_1}$ and ${T_2}$ respectively after two games.

$\,\,\,\,P\,\left( {X > Y} \right)$ is

A

${1 \over 4}$

B

${5 \over 12}$

C

${1 \over 2}$

D

${7 \over 12}$

Question 17: Palette 17/84

JEE Advanced 2016 MCQ

Football teams ${T_1}$ and ${T_2}$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of ${T_1}$ winning, drawing and losing a game against ${T_2}$ are ${1 \over 2},{1 \over 6}$ and ${1 \over 3}$ respectively. Each team gets $3$ points for a win, $1$ point for a draw and $0$ point for a loss in a game. Let $X$ and $Y$ denote the total points scored by teams ${T_1}$ and ${T_2}$ respectively after two games.

$P\,\left( {X = Y} \right)$ is

A

${{11} \over {36}}$

B

${{1} \over {3}}$

C

${{13} \over {36}}$

D

${{1} \over {2}}$

Question 18: Palette 18/84

JEE Advanced 2016 MCQ

A computer producing factory has only two plants ${T_1}$ and ${T_2}.$ Plant ${T_1}$ produces $20$% and plant ${T_2}$ produces $80$% of the total computers produced. $7$% of computers produced in the factory turn out to be defective. It is known that $P$ (computer turns out to be defective given that it is produced in plant ${T_1}$)
$ = 10P$ (computer turns out to be defective given that it is produced in plant ${T_2}$),
where $P(E)$ denotes the probability of an event $E$. A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant ${T_2}$ is

A

${{36} \over {73}}$

B

${{47} \over {79}}$

C

${{78} \over {93}}$

D

${{75} \over {83}}$

Question 19: Palette 19/84

JEE Advanced 2015 MSQ

Let ${n_1}$ and ${n_2}$ be the number of red and black balls, respectively, in box ${\rm I}$. Let ${n_3}$ and ${n_4}$ be the number of red and black balls, respectively, in box ${\rm I}{\rm I}.$

A ball is drawn at random from box ${\rm I}$ and transferred to box ${\rm I}$${\rm I}.$ If the probability of drawing a red ball from box ${\rm I},$ after this transfer, is ${1 \over 3},$ then the correct option(s) with the possible values of ${n_1}$ and ${n_2}$ is(are)

A

${n_1} = 4$ and ${n_2} = 6$

B

${n_1} = 2$ and ${n_2} = 3$

C

${n_1} = 10$ and ${n_2} = 20$

D

${n_1} = 3$ and ${n_2} = 6$

Question 20: Palette 20/84

JEE Advanced 2015 MSQ

Let ${n_1}$ and ${n_2}$ be the number of red and black balls, respectively, in box ${\rm I}$. Let ${n_3}$ and ${n_4}$ be the number of red and black balls, respectively, in box ${\rm I}{\rm I}.$

One of the two boxes, box ${\rm I}$ and box ${\rm I}{\rm I},$ was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probability that this red ball was drawn from box ${\rm I}{\rm I}$ is ${1 \over 3},$ then the correct option(s) with the possible values of ${n_1}$ ${n_2},$ ${n_3}$ and ${n_4}$ is (are)

A

${n_1} = 3,{n_2} = 3,{n_3} = 5,{n_4} = 15$

B

${n_1} = 3,{n_2} = 6,{n_3} = 10,{n_4} = 50$

C

${n_1} = 8,{n_2} = 6,{n_3} = 5,{n_4} = 20$

D

${n_1} = 6,{n_2} = 12,{n_3} = 5,{n_4} = 20$

Question 21: Palette 21/84

JEE Advanced 2014 MCQ

Box $1$ contains three cards bearing numbers $1,2,3;$ box $2$ contains five cards bearing numbers $1,2,3,4,5;$ and box $3$ contains seven cards bearing numbers $1,2,3,4,5,6,7.$ A card is drawn from each of the boxes. Let ${x_i}$ be number on the card drawn from the ${i^{th}}$ box, $i=1,2,3.$

The probability that ${x_1} + {x_2} + {x_3}$ is odd, is

A

${{29} \over {105}}$

B

${{53} \over {105}}$

C

${{57} \over {105}}$

D

${{1} \over {2}}$

Question 22: Palette 22/84

JEE Advanced 2014 MCQ

Box $1$ contains three cards bearing numbers $1,2,3;$ box $2$ contains five cards bearing numbers $1,2,3,4,5;$ and box $3$ contains seven cards bearing numbers $1,2,3,4,5,6,7.$ A card is drawn from each of the boxes. Let ${x_i}$ be number on the card drawn from the ${i^{th}}$ box, $i=1,2,3.$

The probability that ${x_1},$, ${x_2},$ ${x_3}$ are in an arithmetic progression, is

A

${{9} \over {105}}$

B

${{10} \over {105}}$

C

${{11} \over {105}}$

D

${{7} \over {105}}$

Question 23: Palette 23/84

JEE Advanced 2014 MCQ

Three boys and two girls stand in a queue. The probability, that the number of boys ahead of every girl is at least one more than the number of girls ahead of her, is

A

${1 \over 2}$

B

${1 \over 3}$

C

${2 \over 3}$

D

${3 \over 4}$

Question 24: Palette 24/84

JEE Advanced 2013 MCQ

A box ${B_1}$ contains $1$ white ball, $3$ red balls and $2$ black balls. Another box ${B_2}$ contains $2$ white balls, $3$ red balls and $4$ black balls. A third box ${B_3}$ contains $3$ white balls, $4$ red balls and $5$ black balls.

If $2$ balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the other ball is red, the probability that these $2$ balls are drawn from box ${B_2}$ is

A

${{116} \over {181}}$

B

${{126} \over {181}}$

C

${{65} \over {181}}$

D

${{55} \over {181}}$

Question 25: Palette 25/84

JEE Advanced 2013 MCQ

A box ${B_1}$ contains $1$ white ball, $3$ red balls and $2$ black balls. Another box ${B_2}$ contains $2$ white balls, $3$ red balls and $4$ black balls. A third box ${B_3}$ contains $3$ white balls, $4$ red balls and $5$ black balls.

If $1$ ball is drawn from each of the boxex ${B_1},$ ${B_2}$ and ${B_3},$ the probability that all $3$ drawn balls are of the same colour is

A

${{82} \over {648}}$

B

${{90} \over {648}}$

C

${{558} \over {648}}$

D

${{566} \over {648}}$

Question 26: Palette 26/84

JEE Advanced 2013 MCQ

Four persons independently solve a certain problem correctly with probabilities ${1 \over 2},{3 \over 4},{1 \over 4},{1 \over 8}.$ Then the probability that the problem is solved correctly by at least one of them is

A

${{235} \over {256}}$

B

${{21} \over {256}}$

C

${{3} \over {256}}$

D

${{253} \over {256}}$

Question 27: Palette 27/84

JEE Advanced 2012 MCQ

Four fair dice ${D_1,}$ ${D_2,}$ ${D_3}$ and ${D_4}$ ; each having six faces numbered $1, 2, 3, 4, 5$ and $6$ are rolled simultaneously. The probability that ${D_4}$ shows a number appearing on one of ${D_1},$ ${D_2}$ and ${D_3}$ is

A

${{91} \over {216}}$

B

${{108} \over {216}}$

C

${{125} \over {216}}$

D

${{127} \over {216}}$

Question 28: Palette 28/84

JEE Advanced 2012 MSQ

Let $X$ and $Y$ be two events such that $P\left( {X|Y} \right) = {1 \over 2},$ $P\left( {Y|X} \right) = {1 \over 3}$ and $P\left( {X \cap Y} \right) = {1 \over 6}.$ Which of the following is (are) correct ?

A

$P\left( {X \cup Y} \right) = {2 \over 3}$

B

$X$ and $Y$ are independent

C

$X$ and $Y$ are not independent

D

$P\left( {{X^c} \cap Y} \right) = {1 \over 3}$

Question 29: Palette 29/84

JEE Advanced 2012 MSQ

A ship is fitted with three engines ${E_1},{E_2}$ and ${E_3}$. The engines function independently of each other with respective probabilities ${1 \over 2},{1 \over 4}$ and ${1 \over 4}$. For the ship to be operational at least two of its engines must function. Let $X$ denote the event that the ship is operational and Let ${X_1},{X_2}$ and ${X_3}$ denote respectively the events that the engines ${E_1},{E_2}$ and ${E_3}$ are functioning. Which of the following is (are) true?

A

$P\left[ {X_1^c|X} \right] = {3 \over {16}}$

B

$P$ [exactly two engines of the ship are functioning $\left. {|X} \right] = {7 \over 8}$

C

$P\left[ {X|{X_2}} \right] = {5 \over {16}}$

D

$P\left[ {X|{X_1}} \right] = {7 \over {16}}$

Question 30: Palette 30/84

JEE Advanced 2011 MCQ

Given that the drawn ball from ${U_2}$ is white, the probability that head appeared on the coin is

A

${{17} \over {23}}$

B

${{11} \over {23}}$

C

${{15} \over {23}}$

D

${{12} \over {23}}$

Question 31: Palette 31/84

JEE Advanced 2011 MCQ

The probability of the drawn ball from ${U_2}$ being white is

A

${{13} \over {30}}$

B

${{23} \over {30}}$

C

${{19} \over {30}}$

D

${{11} \over {30}}$

Question 32: Palette 32/84

JEE Advanced 2011 MSQ

Let $E$ and $F$ be two independent events. The probability that exactly one of them occurs is $\,{{11} \over {25}}$ and the probability of none of them occurring is $\,{{2} \over {25}}$. If $P(T)$ denotes the probability of occurrence of the event $T,$ then

A

$P\left( E \right) = {4 \over 5},P\left( F \right) = {3 \over 5}$

B

$P\left( E \right) = {1 \over 5},P\left( F \right) = {2 \over 5}$

C

$P\left( E \right) = {2 \over 5},P\left( F \right) = {1 \over 5}$

D

$P\left( E \right) = {3 \over 5},P\left( F \right) = {4 \over 5}$

Question 33: Palette 33/84

JEE Advanced 2010 MCQ

Let $\omega $ be a complex cube root of unity with $\omega \ne 1.$ A fair die is thrown three times. If ${r_1},$ ${r_2}$ and ${r_3}$ are the numbers obtained on the die, then the probability that ${\omega ^{{r_1}}} + {\omega ^{{r_2}}} + {\omega ^{{r_3}}} = 0$ is

A

${1 \over 18}$

B

${1 \over 9}$

C

${2 \over 9}$

D

${1 \over 36}$

Question 34: Palette 34/84

JEE Advanced 2010 MCQ

A signal which can be green or red with probability ${4 \over 5}$ and ${1 \over 5}$ respectively, is received by station A and then transmitted to station $B$. The probability of each station receving the signal correctly is ${3 \over 4}$. If the signal received at atation $B$ is green, then the probability that the original signal was green is

A

${3 \over 5}$

B

${6 \over 7}$

C

${20 \over 23}$

D

${9 \over 20}$

Question 35: Palette 35/84

JEE Advanced 2009 MCQ

The probability that X = 3 equals

A

${{25} \over {216}}$

B

${{25} \over {36}}$

C

${{5} \over {36}}$

D

${{125} \over {216}}$

Question 36: Palette 36/84

JEE Advanced 2009 MCQ

The probability that $X\ge3$ equals :

A

${{125} \over {216}}$

B

${{25} \over {36}}$

C

${{5} \over {36}}$

D

${{25} \over {216}}$

Question 37: Palette 37/84

JEE Advanced 2009 MCQ

The conditional probability that $X\ge6$ given $X>3$ equals :

A

${{125} \over {216}}$

B

${{25} \over {216}}$

C

${{5} \over {36}}$

D

${{25} \over {36}}$

Question 38: Palette 38/84

JEE Advanced 2008 MCQ

An experiment has 10 equally likely outcomes. Let A and B be two non-empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent is :

A

2, 4 or 8

B

3, 6 or 9

C

4 or 8

D

5 or 10

Question 39: Palette 39/84

JEE Advanced 2008 MCQ

Consider the system of equations $ax+by=0; cx+dy=0,$
where $a,b,c,d$ $ \in \left\{ {0,1} \right\}$

STATEMENT - 1 : The probability that the system of equations has a unique solution is ${3 \over 8}.$ and

STATEMENT - 2 : The probability that the system of equations has a solution is $1.$

A

STATEMENT - 1 is True, STATEMENT - 2 is True; STATEMENT - 2 is a correct explanation for STATEMENT - 1

B

STATEMENT - 1 is True, STATEMENT - 2 is True; STATEMENT - 2 is NOT a correct explanation for STATEMENT - 1

C

STATEMENT - 1 is True, STATEMENT - 2 is False.

D

STATEMENT - 1 is False, STATEMENT - 2 is True.

Question 40: Palette 40/84

JEE Advanced 2007 MCQ

Let ${H_1},{H_2},....,{H_n}$ be mutually exclusive and exhaustive events with $P\left( {{H_1}} \right) > 0,i = 1,2,.....,n.$ Let $E$ be any other event with $0 < P\left( E \right) < 1.$
STATEMENT-1:
$P\left( {{H_1}|E} \right) > P\left( {E|{H_1}} \right).P\left( {{H_1}} \right)$ for $i=1,2,....,n$ because

STATEMENT-2: $\sum\limits_{i = 1}^n {P\left( {{H_i}} \right)} = 1.$

A

Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

B

Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.

C

Statement-1 is True, Statement-2 is False.

D

Statement-1 is False, Statement-2 is True

Question 41: Palette 41/84

JEE Advanced 2007 MCQ

One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife is

A

${1 \over 2}$

B

${1 \over 3}$

C

${2 \over 5}$

D

${1 \over 5}$

Question 42: Palette 42/84

JEE Advanced 2007 MCQ

Let ${E^c}$ denote the complement of an event $E.$ Let $E, F, G$ be pairwise independent events with $P\left( G \right) > 0$ and $P\left( {E \cap F \cap G} \right) = 0.$ Then $P\left( {{E^c} \cap {F^c}|G} \right)$ equals

A

$P\left( {{E^c}} \right) + P\left( {{F^c}} \right)$

B

$P\left( {{E^c}} \right) - P\left( {{F^c}} \right)$

C

$P\left( {{E^c}} \right) - P\left( F \right)$

D

$P\left( E \right) - P\left( {{F^c}} \right)$

Question 43: Palette 43/84

JEE Advanced 2007 MCQ

One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife is

A

$\frac{1}{2}$

B

$\frac{1}{3}$

C

$\frac{2}{5}$

D

$\frac{1}{5}$

Question 44: Palette 44/84

JEE Advanced 2007 MCQ

Let H$_1$, H$_2$, ..., H$_n$ be mutually exclusive and exhaustive events with P(H$_i$) > 0, i = 1, 2, ..., n. Let E be any other event with 0 < P(E) < 1.

Statement 1 : P(H$_i$ | E) > P(E | H$_i$). P(H$_i$) for $i=1,2,...,n$.

Statement 2 : $\sum\limits_{i = 1}^n {P({H_i}) = 1} $.

A

Statement 1 is True, Statement 2 is True, Statement 2 is a CORRECT explanation for Statement 1

B

Statement 1 is True, Statement 2 is True, Statement 2 is NOT a CORRECT explanation for Statement 1

C

Statement 1 is True, Statement 2 is False

D

Statement 1 is False, Statement 2 is True

Question 45: Palette 45/84

JEE Advanced 2006 MCQ

If $\mathrm{P}\left(u_{i}\right) \propto i$, where $i=1,2,3, \ldots n$, then $\lim_\limits{n \rightarrow \infty} \mathrm{P}(w)$ is equal to:

A

1

B

$\frac{2}{3}$

C

$\frac{3}{4}$

D

$\frac{1}{4}$

Question 46: Palette 46/84

JEE Advanced 2006 MCQ

If $\mathrm{P}\left(u_{i}\right)=c$, where $c$ is a constant then $\mathrm{P}\left(u_{n} / w\right)$ is equal to:

A

$\frac{2}{n+1}$

B

$\frac{1}{n+1}$

C

$\frac{n}{n+1}$

D

$\frac{1}{2}$

Question 47: Palette 47/84

JEE Advanced 2006 MCQ

If $n$ is even and E denotes the event of choosing even numbered urn $\left(\mathrm{P}\left(u_{i}\right)=\frac{1}{n}\right)$, then the value of $\mathrm{P}(w / \mathrm{E})$ is :

A

$\frac{n+2}{2 n+1}$

B

$\frac{n+2}{2(n+1)}$

C

$\frac{n}{n+1}$

D

$\frac{1}{n+1}$

Question 48: Palette 48/84

JEE Advanced 2005 MCQ

A six faced fair dice is thrown until $1$ comes, then the probability that $1$ comes in even no. of trials is

A

$5/11$

B

$5/6$

C

$6/11$

D

$1/6$

Question 49: Palette 49/84

JEE Advanced 2005 MCQ

A person goes office either by car, scooter, bus or train, proability of which being $\frac{1}{7}, \frac{3}{2}, \frac{2}{7}$ and $\frac{1}{7}$, respectively. Probability that he reaches office late, if he takes car, scooter, bus or train is $\frac{2}{9}, \frac{1}{9}, \frac{4}{9}$ and $\frac{1}{9}$, respectively. Given that he reached office in time, then what is the probability that he travelled by a car?

A

$\frac{1}{7}$

B

$\frac{1}{8}$

C

$\frac{3}{7}$

D

$\frac{3}{8}$

Question 50: Palette 50/84

JEE Advanced 2004 MCQ

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

$4/1155$

Question 51: Palette 51/84

JEE Advanced 2003 MCQ

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

$4/5$

Question 52: Palette 52/84

JEE Advanced 2003 MCQ

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

$1/9$

Question 53: Palette 53/84

JEE Advanced 1999 MCQ

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

$1/49$

Question 54: Palette 54/84

JEE Advanced 1999 MSQ

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

$pmc=1/4$

Question 55: Palette 55/84

JEE Advanced 1998 MCQ

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

$1/3$

Question 56: Palette 56/84

JEE Advanced 1998 MCQ

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

$1/4$

Question 57: Palette 57/84

JEE Advanced 1998 MCQ

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

$1/5$

Question 58: Palette 58/84

JEE Advanced 1998 MCQ

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

Question 59: Palette 59/84

JEE Advanced 1998 MCQ

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

$3/16$

Question 60: Palette 60/84

JEE Advanced 1998 MSQ

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

$P\left( {E/\overline F } \right) + P\left( {\overline E /\overline F } \right) = 1$

Question 61: Palette 61/84

JEE Advanced 1996 MCQ

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

${{3p + 2{p^2}} \over 4}$

Question 62: Palette 62/84

JEE Advanced 1995 MCQ

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

$2/3$

Question 63: Palette 63/84

JEE Advanced 1995 MCQ

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

$1/20$

Question 64: Palette 64/84

JEE Advanced 1995 MSQ

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

$P\left( {A/B} \right) = P\left( A \right)$

Question 65: Palette 65/84

JEE Advanced 1994 MCQ

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

Question 66: Palette 66/84

JEE Advanced 1993 MCQ

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

$65/81$

Question 67: Palette 67/84

JEE Advanced 1993 MSQ

$E$ 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

$\,P\left( E \right) = 1/4,P\left( F \right) = 1/3$

Question 68: Palette 68/84

JEE Advanced 1992 MCQ

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

$0.0250$

Question 69: Palette 69/84

JEE Advanced 1991 MSQ

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.

Question 70: Palette 70/84

JEE Advanced 1989 MSQ

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

$P\left( {E|F} \right) + P\left( {{E^c}|F} \right) = 1.$

Question 71: Palette 71/84

JEE Advanced 1989 MCQ

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

Question 72: Palette 72/84

JEE Advanced 1988 MCQ

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

$51/101.$

Question 73: Palette 73/84

JEE Advanced 1988 MSQ

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

$P\left( A \right) + P\left( B \right) + P\left( {A \cup B} \right)\,\,$

Question 74: Palette 74/84

JEE Advanced 1986 MCQ

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

$p = 1,q = {1 \over 2}$

Question 75: Palette 75/84

JEE Advanced 1986 MCQ

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

$1.4$

Question 76: Palette 76/84

JEE Advanced 1984 MCQ

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

$1/2$

Question 77: Palette 77/84

JEE Advanced 1984 MCQ

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

$3/28$

Question 78: Palette 78/84

JEE Advanced 1984 MSQ

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

$P\left( {M \cap {N^c}} \right) + P\left( {{M^c} \cap N} \right)$

Question 79: Palette 79/84

JEE Advanced 1983 MCQ

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

Question 80: Palette 80/84

JEE Advanced 1983 MCQ

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

Question 81: Palette 81/84

JEE Advanced 1982 MCQ

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).

Question 82: Palette 82/84

JEE Advanced 1980 MCQ

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

Question 83: Palette 83/84

JEE Advanced 1980 MCQ

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

Question 84: Palette 84/84

JEE Advanced 1979 MCQ

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.