## MCQ (Single Correct Answer)

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...

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 fac...

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 C1 and C2 be two biased coins such that the probabilities of getting head in a single toss are $${{2 \over 3}}$$ and $${{1 \over 3}}$$, respective...

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 initia...

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 initia...

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

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. Th...

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. Th...

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...

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...

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 ca...

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 ca...

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 an...

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 an...

Four persons independently solve a certain problem correctly with probabilities $${1 \over 2},{3 \over 4},{1 \over 4},{1 \over 8}.$$ Then the probabil...

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. T...

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

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

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 th...

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...

A fair die is tossed repeatedly until a six is obtained. Let $$X$$ denote the number of tosses required.
The probability that $$X=3$$ equals

A fair die is tossed repeatedly until a six is obtained. Let $$X$$ denote the number of tosses required.
The probability that $$X \ge 3$$ equals...

A fair die is tossed repeatedly until a six is obtained. Let $$X$$ denote the number of tosses required.
The conditional probability that $$X \ge 6$$...

An experiment has $$10$$ equally likely outcomes. Let $$A$$ and $$B$$ be non-empty events of the experiment. If $$A$$ consists of $$4$$ outcomes, the ...

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 sys...

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( ...

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 m...

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 o...

There are $$n$$ urns, each of these contain $$n+1$$ balls. The ith urn contains $$i$$ white balls and $$(n+1-i)$$ red balls. Let $${u_i}$$ be the even...

There are $$n$$ urns, each of these contain $$n+1$$ balls. The ith urn contains $$i$$ white balls and $$(n+1-i)$$ red balls. Let $${u_i}$$ be the even...

There are $$n$$ urns, each of these contain $$n+1$$ balls. The ith urn contains $$i$$ white balls and $$(n+1-i)$$ red balls. Let $${u_i}$$ be the even...

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

If three distinct numbers are chosen randomly from the first $$100$$ natural numbers, then the probability that all three of them are divisible by bot...

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 ...

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...

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 divisi...

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 dr...

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

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 machi...

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 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

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 $$C$$ occu...

Three of six vertices of a regular hexagon are chosen at random. The probability that the triangle with three vertices is equilateral, equals

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...

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 indepen...

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 fac...

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...

One hundred identical coins, each with probability, $$p,$$ of showing up heads are tossed once. If $$0 < p < 1$$ and the probability of heads sh...

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 $$I...

The probability that at least one of the events $$A$$ and $$B$$ occurs is $$0.6$$. If $$A$$ and $$B$$ occur simultaneously with probability $$0.2,$$ t...

Three identical dice are rolled. The probability that the same number will appear on each of them is

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 re...

Fifteen coupons are numbered $$1, 2 ........15,$$ respectively. Seven coupons are selected at random one at a time with replacement. The probability t...

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 {\ove...

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 pr...

Two events $$A$$ and $$B$$ have probabilities $$0.25$$ and $$0.50$$ respectively. The probability that both $$A$$ and $$B$$ occur simultaneously is $$...

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 numbe...

## Numerical

In a study about a pandemic, data of 900 persons was collected. It was found that
190 persons had symptom of fever,
220 persons had symptom of cough...

A number of chosen at random from the set {1, 2, 3, ....., 2000}. Let p be the probability that the chosen number is a multiple of 3 or a multiple of ...

Three numbers are chosen at random, one after another with replacement, from the set S = {1, 2, 3, ......, 100}. Let p1 be the probability that the ma...

Three numbers are chosen at random, one after another with replacement, from the set S = {1, 2, 3, ......, 100}. Let p1 be the probability that the ma...

The probability that a missile hits a target successfully is 0.75. In order to destroy the target completely, at least three successful hits are requi...

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 is observed. This process is ...

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 given byE1 = {A$$ \in $$S : d...

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least $$0.96,$$ is

Of the three independent events $${E_1},{E_2}$$ and $${E_3},$$ the probability that only $${E_1}$$ occurs is $$\alpha ,$$ only $${E_2}$$ occurs is $$\...

## MCQ (More than One Correct Answer)

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...

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 gree...

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

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 ...

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 ...

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} ...

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...

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 ...

The probabilities that a student passes in Mathematics, Physics and Chemistry are $$m, p$$ and $$c,$$ respectively. Of these subjects, the student has...

If $$\overline E $$ and $$\overline F $$ are the complementary events of events $$E$$ and $$F$$ respectively and if $$0 < P\left( F \right) < 1...

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\...

$$E$$ and $$F$$ are two independent events. The probability that both $$E$$ and $$F$$ happen is $$1/12$$ and the probability that neither $$E$$ nor $$...

For any two events $$A$$ and $$B$$ in a simple space

If $$E$$ and $$F$$ are independent events such that $$0 < P\left( E \right) < 1$$ and $$0 < P\left( F \right) < 1,$$ then

For two given events $$A$$ and $$B,$$ $$P\left( {A \cap B} \right)$$

If $$M$$ and $$N$$ are any two events, the probability that exactly one of them occurs is

## Subjective

A person goes to office either by car, scooter, bus or train, the probability of which being $${1 \over 7},{3 \over 7},{2 \over 7}$$ and $${1 \over 7}...

$$A$$ 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...

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 ...

$$A$$ 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}...

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...

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 $$1/2$$, ...

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 sam...

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...

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) consecu...

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 $...

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 pr...

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...

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 ...

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 ...

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 ...

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 i...

A lot contains $$50$$ defective and $$50$$ non defective bulbs. Two bulbs are drawn at random, one at a time, with replacement. The events $$A, B, C$$...

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 gues...

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...

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 $$5$$...

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 pais...

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 ...

A lot contains $$20$$ articles. The probability that the lot contains exactly $$2$$ defective articles is $$0.4$$ and the probability that the lot con...

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...

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...

$$A, B, C$$ 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...

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...

$$A$$ and $$B$$ are two candidates seeking admission in $$IIT.$$ The probability that $$A$$ is selected is $$0.5$$ and the probability that both $$A$$...

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, se...

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....

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 t...

## Fill in the Blanks

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}} \ri...

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 bl...

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 on...

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 i...

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 ...........

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 $$A$$ ...

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 ...

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 ...

$$P\left( {A \cup B} \right) = P\left( {A \cap B} \right)$$ if and only if the relation between $$P(A)$$ and $$P(B)$$ is .............

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 f...

## True or False

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...

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 $$1/35$$