43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Physics

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Chapter 27

Probability - all with Video Answers

Educators

Chapter Questions

Problem 1

In a game two players $A$ and $B$ take turns in throwing a pair of fair dice starting with player $A$ and total of scores on the two dice, in each throw is noted. $A$ wins the game if he throws a total of 6 before $B$ throws a total of 7 and $B$ wins the game if he throws a total of 7 before $A$ throws a total of six. The game stops as soon as either of the players wins. The probability of $A$ winning the game is :
[Main Sep. 04, 2020 (II)]
(a) $\frac{5}{31}$
(b) $\frac{31}{61}$
(c) $\frac{5}{6}$
(d) $\frac{30}{61}$

Aakash Goyal

Aakash Goyal

Numerade Educator

Problem 2

A die is thrown two times and the sum of the scores appearing on the die is observed to be a multiple of 4 . Then the conditional probability that the score 4 has appeared atleast once is :
[Main Sep. 03, 2020 (I)](a) $\frac{1}{4}$
(b) $\frac{1}{3}$
(c) $\frac{1}{8}$
(d) $\frac{1}{9}$

Aakash Goyal

Numerade Educator

Problem 3

The probability that a randomly chosen 5 -digit number is made from exactly two digits is :
[Main Sep. 03, $\mathbf{2 0 2 0}$ (II)]
(a) $\frac{135}{10^{4}}$
(b) $\frac{121}{10^{4}}$
(c) $\frac{150}{10^{4}}$
(d) $\frac{134}{10^{4}}$

Aman Gupta

Numerade Educator

Problem 4

Box I contains 30 cards numbered 1 to 30 and Box II contains 20 cards numbered 31 to 50 . A box is selected at random and a card is drawn from it. The number on the card is found to be a non-prime number. The probability that the card was drawn from Box I is :
[Main Sep. 02, 2020 (I)]
(a) $\frac{2}{3}$
(b) $\frac{8}{17}$
(c) $\frac{4}{17}$
(d) $\frac{2}{5}$

Aman Gupta

Numerade Educator

Problem 5

Let $\mathrm{E}^{\mathrm{C}}$ denote the complement of an event $\mathrm{E}$. Let $\mathrm{E}_{1}, \mathrm{E}_{2}$ and $\mathrm{E}_{3}$ be any pairwise independent events with $\mathrm{P}\left(\mathrm{E}_{1}\right)>0$
and
$\mathrm{P}\left(\mathrm{E}_{1} \cap \mathrm{E}_{2} \cap \mathrm{E}_{3}\right)=0$
Then $\mathrm{P}\left(\mathrm{E}_{2}^{\mathrm{C}} \cap \mathrm{E}_{3}^{\mathrm{C}} / \mathrm{E}_{1}\right)$ is equal to :
[Main Sep. 02, 2020 (II)]
(a) $\mathrm{P}\left(\mathrm{E}_{2}^{\mathrm{C}}\right)+\mathrm{P}\left(\mathrm{E}_{3}\right)$
(b) $\mathrm{P}\left(\mathrm{E}_{3}^{\mathrm{C}}\right)-\mathrm{P}\left(\mathrm{E}_{2}^{\mathrm{C}}\right)$
(c) $\mathrm{P}\left(\mathrm{E}_{3}\right)-\mathrm{P}\left(\mathrm{E}_{2}^{\mathrm{C}}\right)$
(d) $\mathrm{P}\left(\mathrm{E}_{3}^{\mathrm{C}}\right)-\mathrm{P}\left(\mathrm{E}_{2}\right)$

Aman Gupta

Numerade Educator

Problem 6

In a box, there are 20 cards, out of which 10 are labelled as $\mathrm{A}$ and the remaining 10 are labelled as B. Cards are drawn at random, one after the other and with replacement, till a second A-card is obtained. The probability that the second A-card appears before the third B-card is :
[Main Jan. 9, 2020 (I)]
(a) $\frac{9}{16}$
(b) $\frac{11}{16}$
(c) $\frac{13}{16}$
(d) $\frac{15}{16}$

Aman Gupta

Numerade Educator

Problem 7

Let $A$ and $B$ be two independent events such that $P(A)=\frac{1}{3}$ and $P(B)$ $=\frac{1}{6}$. Then, which of the following is TRUE ? [Main Jan. 8, 2020 (I)]
(a) $P(A / B)=\frac{2}{3}$
(b) $P\left(A / B^{\prime}\right)=\frac{1}{3}$
(c) $P\left(A^{\prime} / B^{\prime}\right)=\frac{1}{3}$
(d) $P(A /(A \square B))=\frac{1}{4}$

Aman Gupta

Numerade Educator

Problem 8

An unbiased coin is tossed 5 times. Suppose that a variable $X$ is assigned the value $k$ when $k$ consecutive heads are obtained for $k=3$, 4,5, otherwise $X$ takes the value $-1$. Then the expected value of $X$, is:
[Main Jan. 7, $\mathbf{2 0 2 0}(\mathbf{I})]$
(a) $\frac{3}{16}$
(b) $\frac{1}{8}$
(c) $-\frac{3}{16}$
(d) $-\frac{1}{8}$

Aman Gupta

Numerade Educator

Problem 9

In a workshop, there are five machines and the probability of any one of them to be out of service on a day is $\frac{1}{4} .$ If the probability that at most two machines will be out of service on the same day is $\left(\frac{3}{4}\right)^{3} k$, then $k$ is equal to:
[Main Jan. 7, 2020 (II)]
(a) $\frac{17}{8}$
(b) $\frac{17}{4}$
(c) $\frac{17}{2}$
(d) 4

Aman Gupta

Numerade Educator

Problem 10

Let $C_{1}$ and $C_{2}$ be two biased coins such that the probabilities of getting head in a single toss are $\frac{2}{3}$ and $\frac{1}{3}$, respectively. Suppose $\alpha$ is the number of heads that appear when $C_{1}$ is tossed twice, independently, and suppose $\beta$ is the number of heads that appear when $C_{2}$ is tossed twice, independently. Then the probability that the roots of the quadratic polynomial $x^{2}-\alpha x+\beta$ are real and equal, is [Adv. 2020]
(a) $\frac{40}{81}$
(b) $\frac{20}{81}$
(c) $\frac{1}{2}$
(d) $\frac{1}{4}$

Aman Gupta

Numerade Educator

Problem 11

Minimum number of times a fair coin must be tossed so that the probability of getting at least one head is more than $99 \%$ is :
[Main April 10, 2019 (II)]
(a) 5
(b) 6
(c) 8
(d) 7

Aman Gupta

Numerade Educator

Problem 12

Let $\mathrm{A}$ and $\mathrm{B}$ be two non-null events such that $\mathrm{A} \subset \mathrm{B}$. Then, which of the following statements is always correct? [Main April $\mathbf{0 8 , 2 0 1 9}(\mathbf{I})]$
(a) $\mathrm{P}(\mathrm{A} \mid \mathrm{B})=\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A})$
(b) $\mathrm{P}(\mathrm{A} \mid \mathrm{B}) \geq \mathrm{P}(\mathrm{A})$
(c) $\mathrm{P}(\mathrm{A} \mid \mathrm{B}) \leq \mathrm{P}(\mathrm{A})$
(d) $\mathrm{P}(\mathrm{A} \mid \mathrm{B})=1$

Aman Gupta

Numerade Educator

Problem 13

If the probability of hitting a target by a shooter, in any shot, is $\frac{1}{3}$,
then the minimum number of independent shots at the target required by him so that the probability of hitting the target at least once is greater than $\frac{5}{6}$, is:
[Main Jan. 10, 2019 (II)]
(a) 3
(b) 6
(c) 5
(d) 4

Aman Gupta

Numerade Educator

Problem 14

An urn contains 5 red and 2 green balls. A ball is drawn at random from the urn. If the drawn ball is green, then a red ball is added to the urn and if the drawn ball is red, then a green ball is added to the urn; the original ball is not returned to the urn. Now, a second ball is drawn at random from it. The probability that the second ball is red is:
[Main Jan. 09, $\mathbf{2 0 1 9}$ (II)]
(a) $\frac{21}{49}$
(b) $\frac{27}{49}$
(c) $\frac{26}{49}$
(d) $\frac{32}{49}$

Aman Gupta

Numerade Educator

Problem 15

If two different numbers are taken from the set $(0,1,2,3, \ldots \ldots, 10)$, then the probability that their sum as well as absolute difference are both multiple of 4, is :
[Main 2017]
(a) $\frac{7}{55}$
(b) $\frac{6}{55}$
(c) $\frac{12}{55}$
(d) $\frac{14}{55}$

Aman Gupta

Numerade Educator

Problem 16

An unbiased coin is tossed eight times. The probability of obtaining at least one head and at least one tail is :
[Main Online April 8, 2017]
(a) $\frac{255}{256}$
(b) $\frac{127}{128}$
(c) $\frac{63}{64}$
(d) $\frac{1}{2}$

Aman Gupta

Numerade Educator

Problem 17

Three randomly chosen non-negative integers $\mathrm{x}, \mathrm{y}$ and $\mathrm{z}$ are found to satisfy the equation $\mathrm{x}+\mathrm{y}+\mathrm{z}=10$. Then the probability that $\mathrm{z}$ is even, is [Adv. 2017]
(a) $\frac{36}{55}$
(b) $\frac{6}{11}$
(c) $\frac{1}{2}$
(d) $\frac{5}{11}$

Aman Gupta

Numerade Educator

Problem 18

Let two fair six-faced dice $\mathrm{A}$ and $\mathrm{B}$ be thrown simultaneously. If $\mathrm{E}_{1}$ is the event that die A shows up four, $\mathrm{E}_{2}$ is the event that die $\mathrm{B}$ shows up two and $\mathrm{E}_{3}$ is the event that the sum of numbers on both dice is odd, then which of the following statements is NOT true? [Main 2016]
(a) $\mathrm{E}_{1}$ and $\mathrm{E}_{3}$ are independent.
(b) $\mathrm{E}_{1}, \mathrm{E}_{2}$ and $\mathrm{E}_{3}$ are independent.
(c) $\mathrm{E}_{1}$ and $\mathrm{E}_{2}$ are independent.
(d) $\mathrm{E}_{2}$ and $\mathrm{E}_{3}$ are independent.

Aman Gupta

Numerade Educator

Problem 19

A computer producing factory has only two plants $\mathrm{T}_{1}$ and $\mathrm{T}_{2} .$ Plant $\mathrm{T}_{1}$ produces $20 \%$ and plant $\mathrm{T}_{2}$ produces $80 \%$ of the total computers produced. $7 \%$ of computers produced in the factory turn out to be defective. It is known that
$\mathrm{P}$ (computer turns out to be defective given that it is produced in plant $\mathrm{T}_{1}$ ) $=10 \mathrm{P}$ (computer turns out to be defective given that it is produced in plant $\mathrm{T}_{2}$ ),
where $\mathrm{P}(\mathrm{E})$ denotes the probability of an event $\mathrm{E}$. 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 $\mathrm{T}_{2}$ is [Adv. 2016]
(a) $\frac{36}{73}$
(b) $\frac{47}{79}$
(c) $\frac{78}{93}$
(d) $\frac{75}{83}$

Aman Gupta

Numerade Educator

Problem 20

Let $A$ and $B$ be two events such that $P(\overline{A \cup B})=\frac{1}{6}, P(\overline{A \cap B})=\frac{1}{4}$ and $P(\bar{A})=\frac{1}{4}$, where $\bar{A}$ stands for the complement of the event $\mathrm{A}$. Then the events $\mathrm{A}$ and $\mathrm{B}$ are [Main 2014]
(a) independent but not equally likely.
(b) independent and equally likely.
(c) mutually exclusive and independent.
(d) equally likely but not independent.

Aman Gupta

Numerade Educator

Problem 21

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. Theprobability that $D_{4}$ shows a number appearing on one of $D_{1}, D_{2}$ and $\mathrm{D}_{3}$ is [2012]
(a) $\frac{91}{216}$
(b) $\frac{108}{216}$
(c) $\frac{125}{216}$
(d) $\frac{127}{216}$

Aman Gupta

Numerade Educator

Problem 22

A signal which can be green or red with probability $\frac{4}{5}$ and $\frac{1}{5}$
respectively, is received by station A and then transmitted to station
B. The probability of each station receiving the signal correctly is $\frac{3}{4}$ If the signal received at station $\mathrm{B}$ is green, then the probability that the original signal was green is [2010](a) $\frac{3}{5}$
(b) $\frac{6}{7}$
(c) $\frac{20}{23}$
(d) $\frac{9}{20}$

Aman Gupta

Numerade Educator

Problem 23

Let $\omega$ be a complex cube root of unity with $\omega \neq 1$. 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 $\omega^{r}+\omega^{r_{2}}+\omega^{r_{3}}=0$ is [2010]
(a) $\frac{1}{18}$
(b) $\frac{1}{9}$
(c) $\frac{2}{9}$
(d) $\frac{1}{36}$

Aman Gupta

Numerade Educator

Problem 24

An experiment has 10 equally likely outcomes. Let $A$ and $B$ be nonempty 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 [2008]
(a) 2,4 or 8
(b) 3,6 or 9
(c) 4 or 8
(d) 5 or 10

Aman Gupta

Numerade Educator

Problem 25

Let $E^{c}$ denote the complement of an event $E$. Let $E, F, G$ be pairwise independent events with $P(G)>0$ and $P(E \cap F \cap G)=0$. Then $P\left(E^{c} \cap F^{c} \mid G\right)$ equals
[2007 -3 marks]
(a) $P\left(E^{c}\right)+P\left(F^{c}\right)$
(b) $P\left(E^{c}\right)-P\left(F^{c}\right)$
(c) $P\left(E^{c}\right)-P(F)$
(d) $P(E)-P\left(F^{c}\right)$

Aman Gupta

Numerade Educator

Problem 26

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 [2007-3 marks]
(a) $\frac{1}{2}$
(b) $\frac{1}{3}$
(c) $\frac{2}{5}$
(d) $\frac{1}{5}$

Aman Gupta

Numerade Educator

Problem 27

A six faced fair dice is thrown until 1 comes, then the probability that 1 comes in even no. of trials is [2005S]
(a) $5 / 11$
(b) $5 / 6$
(c) $6 / 11$
(d) $1 / 6$

Aman Gupta

Numerade Educator

Problem 28

Two numbers are selected randomly from the set $S=\{1,2,3,4,5,6\}$ without replacement one by one. The probability that minimum of the two numbers is less than 4 is [2003S]
(a) $1 / 15$
(b) $14 / 15$
(c) $1 / 5$
(d) $4 / 5$

Aman Gupta

Numerade Educator

Problem 29

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 [1999-2 Marks]
(a) $1 / 4$
(b) $1 / 7$
(c) $1 / 8$
(d) $1 / 49$

Aman Gupta

Numerade Educator

Problem 30

Three of the six vertices of a regular hexagon are chosen at random. The probability that the triangle with three vertices is equilateral, equals $[\mathbf{1 9 9 5 S}]$
(a) $1 / 2$
(b) $1 / 5$
(c) $1 / 10$
(d) $1 / 20$

Aman Gupta

Numerade Educator

Problem 31

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, $[1994]$
(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

Aman Gupta

Numerade Educator

Problem 32

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:

Aman Gupta

Numerade Educator

Problem 33

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 [1992 - 2 Marks]
(a) $0.8750$
(b) $0.0875$
(c) $0.0625$
(d) $0.0250$

Aman Gupta

Numerade Educator

Problem 34

Fifteen coupons are numbered $1,2 \ldots .15$,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 [1983-1 Mark]
(a) $\left(\frac{9}{16}\right)^{6}$
(b) $\left(\frac{8}{15}\right)^{7}$
(c) $\left(\frac{3}{5}\right)^{7}$

Aman Gupta

Numerade Educator

Problem 35

If $A$ and $B$ are two events such that $P(A)>0$, and $P(B) \neq 1$, then $P\left(\overline{\frac{A}{\bar{B}}}\right)$ is equal to(a) $1-P\left(\frac{-3}{B}\right)$
(b) $1-P\left(\frac{\bar{A}}{\mathrm{~B}}\right)$
(c) $\frac{1-P(A \cup B)}{P(\bar{B})}$
(d) $\frac{P(\bar{A})}{P(\bar{B})}$
(Here $\bar{A}$ and $\bar{B}$ are complements of $A$ and $B$ respectively).

Aman Gupta

Numerade Educator

Problem 36

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 [1980]
(a) $0.936$
(b) $0.784$
(c) $0.904$
(d) none of these

Aman Gupta

Numerade Educator

Problem 37

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 $[1980]$
(a) $0.39$
(b) $0.25$
(c) $0.11$
(d) none of these

Aman Gupta

Numerade Educator

Problem 38

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 :
[1979]
(a) Mutually exclusive
(b) Independent and mutually exclusive
(c) Dependent
(d) None of these.

Aman Gupta

Numerade Educator

Problem 39

Of the three independent events $\mathrm{E}_{1}, \mathrm{E}_{2}$ and $\mathrm{E}_{3}$, the probability that only $\mathrm{E}_{1}$ occurs is $\alpha$, only $\mathrm{E}_{2}$ occurs is $\beta$ and only $\mathrm{E}_{3}$ occurs is $\gamma$. Let the probability $\mathrm{p}$ that none of events $\mathrm{E}_{1}, \mathrm{E}_{2}$ or $\mathrm{E}_{3}$ occurs satisfy the equations $(\alpha 2 \beta) p=\alpha \beta$ and $(\beta-3 \gamma) p=2 \beta \gamma$. All the given probabilities are assumed to lie in the interval $(0,1)$. [Adv. 2013] Then $\frac{\text { Pr obability of occurrence of } \mathrm{E}_{1}}{\text { Pr obability of occurrence of } \mathrm{E}_{3}}$

Aman Gupta

Numerade Educator

Problem 40

Let $\mathrm{S}$ be the sample space of all $3 \times 3$ matrices with entries from the set $\{0,1\}$. Let the events $E_{1}$ and $E_{2}$ be given by $E_{1}=\{A \in S:$ det $A=0\}$ and $E_{2}=\{A \in S:$ sum of entries of $\mathrm{A}$ is 7$\}$.
a matrix is chosen at random from $\mathrm{S}$, then the conditional probability If $\mathrm{P}\left(E_{1} / E_{2}\right)$ equals

Aman Gupta

Numerade Educator

Problem 41

If two events $A$ and $B$ are such that $P\left(A^{c}\right)=0.3, P(B)=0.4$ and $P$ $\left(A \cap B^{c}\right)=0.5$, then $P\left(B /\left(A \cup B^{c}\right)\right]=\ldots \ldots \ldots \ldots \ldots \ldots$
[1994-2 Marks]

Aman Gupta

Numerade Educator

Problem 42

Let $A$ and $B$ be two events such that $P(A)=0.3$ and $P(A \cup B)=0.8$. If $A$ and $B$ are independent events then $P(B)=\ldots \ldots \ldots \ldots \ldots \ldots$ [1990 - 2 Marks]

Aman Gupta

Numerade Educator

Problem 43

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 $B$. Then 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 ................... [1988 - 2 Marks]

Aman Gupta

Numerade Educator

Problem 44

If $\frac{1+3 p}{3}, \frac{1-p}{4}$ and $\frac{1-2 p}{2}$ are the probabilities of three mutually exclusive events, then the set of all values of $p$ is .................

Aman Gupta

Numerade Educator

Problem 45

A box contains 100 tickets numbered $1,2, \ldots ., 100 .$ 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 $\ldots \ldots \ldots \ldots \ldots$

Aman Gupta

Numerade Educator

Problem 46

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 $0.5$.

Aman Gupta

Numerade Educator

Problem 47

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 $1 / 35$

Aman Gupta

Numerade Educator

Problem 48

There are three bags $\mathrm{B}_{1}, \mathrm{~B}_{2}$, and $\mathrm{B}_{3}$. The bag $\mathrm{B}_{1}$ contains 5 red and 5 green balls, $\mathrm{B}_{2}$ contains 3 red and 5 green balls, and $\mathrm{B}_{3}$ contains $5 \mathrm{red}$ and 3 green balls. Bags $\mathrm{B}_{1}, \mathrm{~B}_{2}$ and $\mathrm{B}_{3}$ have probabilities $\frac{3}{10}, \frac{3}{10}$ and $\frac{4}{10}$ 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? [Adv. 2019]
(a) Probability that the selected bag is $\mathrm{B}_{3}$ and the chosen ball is green equals $\frac{3}{10}$
(b) Probability that the chosen ball is green, given that the selected bag is $\mathrm{B}_{3}$, equals $\frac{3}{8}$
(c) Probability that the selected bag is $\mathrm{B}_{3}$, given that the chosen ball is
green, equals $\frac{ }{13}$
(d) Probability that the chosen ball is green equals $\frac{39}{80}$

Aayush Gupta

Numerade Educator

Problem 49

Let $X$ and $Y$ be two events such that $P(X)=\frac{1}{3}, P(X \mid Y)=\frac{1}{2}$ and $P$ $(\mathrm{Y} \mid \mathrm{X})=\frac{2}{5} .$ Then
[Adv. 2017]
(a) $\mathrm{P}(\mathrm{Y})=\frac{4}{15}$
(b) $\mathrm{P}\left(\mathrm{X}^{\prime} \mid \mathrm{Y}\right)=\frac{1}{2}$
(c) $\mathrm{P}(\mathrm{X} \cap \mathrm{Y})=\frac{1}{5}$
(d) $\mathrm{P}(\mathrm{X} \cup \mathrm{Y})=\frac{2}{5}$

Aman Gupta

Numerade Educator

Problem 50

Let $X$ and $Y$ be two events such that $P(X \mid Y)=\frac{1}{2}, P(Y / X)=\frac{1}{3}$ and $P(X \cap Y)=\frac{1}{6} .$ Which of the following is (are) correct?
(a) $P(X \cup Y)=\frac{2}{3}$
(b) $X$ and $Y$ are independent
(c) $X$ and $Y$ are not independent
(d) $P\left(X^{c} \cap Y\right)=\frac{1}{3}$

Aman Gupta

Numerade Educator

Problem 51

A ship is fitted with three engines $E_{1}, E_{2}$ and $E_{3}$. The engines function independently of each other with respective probabilities $\frac{1}{2}, \frac{1}{4}$ and $\frac{1}{4}$. 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 thatthe engines $E_{1}, E_{2}$ and $E_{3}$ are functioning. Which of the following is(are) true ? [2012]
(a) $P\left[X_{1}^{c} \mid X\right]=\frac{3}{16}$
(b) $P$ [Exactly two engines of the ship are functioning $\mid X]=\frac{7}{8}$
(c) $P\left[X \mid X_{2}\right]=\frac{5}{16}$
(d) $P\left[X \mid X_{1}\right]=\frac{7}{16}$

Muhammad Nawaz

Numerade Educator

Problem 52

Let $E$ and $F$ be two independent events. The probability that exactly one of them occurs is $\frac{11}{25}$ and the probability of none of them
occurring is $\frac{2}{25} .$ If $P(T)$ denotes the probability of occurrence of the event $T$, then(a) $P(E)=\frac{4}{5}, P(F)=\frac{3}{5}$
(b) $P(E)=\frac{1}{5}, P(F)=\frac{2}{5}$
(c) $P(E)=\frac{2}{5}, P(F)=\frac{1}{5}$
(d) $P(E)=\frac{3}{5}, P(F)=\frac{4}{5}$

Aayush Gupta

Numerade Educator

Problem 53

The probabilities that a student passes in Mathematics, Physics and Chemistry are $\mathrm{m}, \mathrm{p}$ and $\mathrm{c}$, 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? [1999-3 Marks]
(a) $p+m+c=19 / 20$
(b) $p+m+c=27 / 20$
(c) $p m c=1 / 10$
(d) $p m c=1 / 4$

Aayush Gupta

Numerade Educator

Problem 54

Seven white balls and three black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals [1998-2 Marks]
(a) $1 / 2$
(b) $7 / 15$
(c) $2 / 15$
(d) $1 / 3$

Aman Gupta

Numerade Educator

Problem 55

If $\mathrm{E}$ and $\mathrm{F}$ are events with $P(E) \leq P(F)$ and $P(E \cap F)>0$, then [1998 - 2 Marks]
(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

Aman Gupta

Numerade Educator

Problem 56

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 [1998-2 Marks]
(a) $1 / 3$
(b) $1 / 6$
(c) $1 / 2$
(d) $1 / 4$

Aman Gupta

Numerade Educator

Problem 57

If $\bar{E}$ and $\bar{F}$ are the complementary events of events $E$ and $\mathrm{F}$ respectively and if $0<P(F)<1$, then [1998 - 2 Marks]
(a) $P(E / F)+P(\bar{E} / F)=1$
(b) $P(E / F)+P(E / \bar{F})=1$
(c) $P(\bar{E} / F)+P(E / \bar{F})=1$
(d) $P(E / \bar{F})+P(\bar{E} / \bar{F})=1$

Aman Gupta

Numerade Educator

Problem 58

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) $3 / 16$

Aman Gupta

Numerade Educator

Problem 59

Let $0<P(A)<1,0<P(B)<1$ and $P(A \cup B)=P(A)+P(B)-P(A) P(B)$ then
(a) $P(B / A)=P(B)-P(A)$
(b) $P\left(A^{\prime}-B^{\prime}\right)=P\left(A^{\prime}\right)-P\left(B^{\prime}\right)$
(c) $P(A \cup B)^{\prime}=P\left(A^{\prime}\right) P\left(B^{\prime}\right)$
(d) $P(A / B)=P(A)$

Aman Gupta

Numerade Educator

Problem 60

$E$ and $F$ are two independent events. The probability that both $\mathrm{E}$ and $\mathrm{F}$ happen is $1 / 12$ and the probability that neither $\mathrm{E}$ nor $\mathrm{F}$ happens is $1 / 2$. Then, [1993 - 2 Marks]
(a) $P(E)=1 / 3, P(F)=1 / 4$
(b) $P(E)=1 / 2, P(F)=1 / 6$
(c) $P(E)=1 / 6, P(F)=1 / 2$
(d) $P(E)=1 / 4, P(F)=1 / 3$

Aman Gupta

Numerade Educator

Problem 61

For any two events $A$ and $B$ in a sample space [1991-2 Marks]
(a) $P(A / B) \geq \frac{P(A)+P(B)-1}{P(B)}, P(B) \neq 0$ is always true
(b) $P(A \cap \bar{B})=P($ A $)-P(A \cap B)$ does not hold
(c) $P(A \cup B)=1-P(\bar{A}) P(\bar{B})$, if $A$ and $B$ are independent
(d) $P(A \cup B)=1-P(\bar{A}) \mathrm{P}(\bar{B})$, if $A$ and $B$ are disjoint.

Aman Gupta

Numerade Educator

Problem 62

If $E$ and $F$ are independent events such that $0<P(E)<1$ and $0<P(F)$ $<1$, then [1989-2 Marks]
(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) $P(E \mid F)+P\left(E^{c} \mid F\right)=1$.

Aman Gupta

Numerade Educator

Problem 63

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 $P(\bar{A})+P$ $(\bar{B})$ is
[1987-2 Marks]
(a) $0.4$
(b) $0.8$
(c) $1.2$
(d) $1.4$
(e) none (Here $\bar{A}$ and $\bar{B}$ are complements of $A$ and $B$, respectively).

Aman Gupta

Numerade Educator

Problem 64

64. 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 the student passing in tests I, II and III are $p, q$ and $\frac{1}{2}$ respectively. If the
probability that the student is successful is $\frac{1}{2}$, then [1986 - 2 Marks]
(a) $p=q=1$
(b) $p=q=\frac{1}{2}$
(c) $p=1, q=0$
(d) $p=1, q=\frac{1}{2}$
(e) none of these

Aman Gupta

Numerade Educator

Problem 65

The probability that, on examination day, the student $S_{1}$ gets the previously allotted seat $\mathrm{R}_{1}$, and NONE of the remaining students gets the seat previously allotted to him/her is [Adv. 2018]
(a) $\frac{3}{40}$
(b) $\frac{1}{8}$
(c) $\frac{7}{40}$
(d) $\frac{1}{5}$

Aman Gupta

Numerade Educator

Problem 66

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 [Adv. 2018]
(a) $\frac{1}{15}$
(b) $\frac{1}{10}$
(c) $\frac{7}{60}$
(d) $\frac{1}{5}$
Passage - 2 Let $\mathrm{n}_{1}$ and $\mathrm{n}_{2}$ be the number of red and black balls, respectively, in box $\mathrm{I}$. Let $\mathrm{n}_{3}$ and $\mathrm{n}_{4}$ be the number of red and black balls, respectively, in box II.

Aman Gupta

Numerade Educator

Problem 67

red. If the probability that this red ball was drawn from box II is $\frac{1}{3}$, 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) $n_{1}=6, n_{2}=12, n_{3}=5, n_{4}=20$One of the two boxes, box I and box II, was selected at random and a ball was drawn randomly out of this box. The ball was found to be

Aman Gupta

Numerade Educator

Problem 68

A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from box I, after this transfer, is $\frac{1}{3}$, 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 $n_{2}=6$
Passage - 3 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 $i^{\text {th }}$ box, $i=1,2,3$.

Aman Gupta

Numerade Educator

Problem 69

The probability that $x_{1}+x_{2}+x_{3}$ is odd, is
(a) $\frac{29}{105}$
(b) $\frac{53}{105}$
(c) $\frac{57}{105}$
(d) $\frac{1}{2}$

Aman Gupta

Numerade Educator

Problem 70

The probability that $x_{1}, x_{2}, x_{3}$ are in an arithmetic progression, is(a) $\frac{9}{105}$
(b) $\frac{10}{105}$
(c) $\frac{11}{105}$
(d) $\frac{7}{105}$

Aman Gupta

Numerade Educator

Problem 71

If 1 ball is drawn from each of the boxes $\mathrm{B}_{1}, \mathrm{~B}_{2}$ and $\mathrm{B}_{3}$, the probability that all 3 drawn balls are of the same colour is [Adv. 2013$]$
(a) $\frac{82}{648}$
(b) $\frac{90}{648}$
(c) $\frac{558}{648}$
(d) $\frac{566}{648}$

Aman Gupta

Numerade Educator

Problem 72

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 $\mathrm{B}_{2}$ is
(a) $\frac{116}{181}$
(b) $\frac{126}{181}$
(c) $\frac{65}{181}$
(d) $\frac{55}{181}$

Aman Gupta

Numerade Educator

Problem 73

The probability of the drawn ball from $U_{2}$ being white is
(a) $\frac{13}{30}$
(b) $\frac{23}{30}$
(c) $\frac{19}{30}$
(d) $\frac{11}{30}$

Aman Gupta

Numerade Educator

Problem 74

Given that the drawn ball from $\mathrm{U}_{2}$ is white, the probability that head appeared on the coin is
(a) $\frac{17}{23}$
(b) $\frac{11}{23}$
(c) $\frac{15}{23}$
(d) $\frac{12}{23}$

Aman Gupta

Numerade Educator

Problem 75

Let $\mathrm{A}$ and $\mathrm{E}$ be any two events with positive probabilities:
Statement - 1: $\mathrm{P}(\mathrm{E} / \mathrm{A}) \geq \mathrm{P}(\mathrm{A} / \mathrm{E}) \mathrm{P}(\mathrm{E})$
Statement - 2: $\mathrm{P}(\mathrm{A} / \mathrm{E}) \geq \mathrm{P}(\mathrm{A} \cap \mathrm{E})$
(a) Both the statements are true
(b) Both the statements are false
(c) Statement- 1 is true, Statement-2 is false
(d) Statement- 1 is false, Statement- 2 is true

Aman Gupta

Numerade Educator

Problem 76

Consider the system of equations $a x+b y=0 ; c x+d y=0$, where $a, b$, $c, d \in\{0,1\}$
STATEMENT - $1:$ The probability that the system of equations has a
and unique solution is $\frac{3}{8}$.
STATEMENT - $2:$ The probability that the system of equations has a solution is 1 . $[2008]$
(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 explaination for STATEMENT - 1
(c) STATEMENT - 1 is True, STATEMENT - 2 is False
(d) STATEMENT - 1 is False, STATEMENT - 2 is True

Aman Gupta

Numerade Educator

Problem 77

Let $H_{1}, H_{2}, \ldots, H_{\mathrm{n}}$ be mutually exclusive and exhaustive events with $P\left(H_{i}\right)>0, i=1,2, \ldots, n$. Let $E$ be any other event with $0<P(E)<1$. STATEMENT-1 :
$P\left(H_{i} \mid E\right)>P\left(E \mid H_{i}\right) \cdot P\left(H_{i}\right)$ for $i=1,2, \ldots, \mathrm{n}$ because
STATEMENT- $2: \sum_{i=1}^{n} P\left(H_{i}\right)=1$.
[2007-3 marks]
(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.

Aman Gupta

Numerade Educator

Problem 78

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

Aman Gupta

Numerade Educator

Problem 79

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)

Aman Gupta

Numerade Educator

Problem 80

$A$ and $B$ are two independent events. $C$ is event in which exactly one of $A$ or $B$ occurs. Prove that
$P(C) \geq P(A \cup B) P(\bar{A} \cap \bar{B})$

Aman Gupta

Numerade Educator

Problem 81

$A$ is targeting to $B, B$ and $C$ are targeting to $A$. Probability of hitting the target by $A, B$ and $C$ are $\frac{2}{3}, \frac{1}{2}$ and $\frac{1}{3}$ respectively. If $A$ is hit then find the probability that $B$ hits the target and $C$ does not.

Aman Gupta

Numerade Educator

Problem 82

For a student to qualify, he must pass at least two out of three exams. The probability that he will pass the $1^{\text {st }}$ exam is $\mathrm{p}$. If he fails in one of
the exams then the probability of his passing in the next exam is $\frac{p}{2}$ otherwise it remains the same. Find the probability that he will qualify.

Aman Gupta

Numerade Educator

Problem 83

A box contains $\mathrm{N}$ coins, $\mathrm{m}$ 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 andthe second time it shows tail. What is the probability that the coin drawn is fair?

Aman Gupta

Numerade Educator

Problem 84

An unbiased die, with faces numbered $1,2,3,4,5,6$, is thrown $\mathrm{n}$ times and the list of $\mathrm{n}$ 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?

Aman Gupta

Numerade Educator

Problem 85

An urn contains $\mathrm{m}$ white and $\mathrm{n}$ black balls. A ball is drawn at random and is put back into the urn along with $\mathrm{k}$ 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?

Aman Gupta

Numerade Educator

Problem 86

A coin has probability $\mathrm{p}$ of showing head when tossed. It is tossed $\mathrm{n}$ 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 $p_{n}=(1-p)$. $p_{n-1}+p(1-p)$ $p_{n-2}$ for all $n \geq 3$.

Aman Gupta

Numerade Educator

Problem 87

Eight players $P_{1}, P_{2}, \ldots \ldots . P_{8}$ play a knock-out tournament. It is known that whenever the players $P_{\mathrm{i}}$ and $P_{\mathrm{j}}$ play, the player $P_{\mathrm{i}}$ 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?

Aman Gupta

Numerade Educator

Problem 88

Three players, $A, B$ and $C$, toss a coin cyclically in that order (that is $A, B, C, A, B, C, A, B, \ldots .)$ till a head shows. Let $\mathrm{p}$ 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 $\beta=(1-p)$ $\alpha$. Determine $\alpha, \beta$ and $\gamma$ (in terms of $p$ ).

Aayush Gupta

Numerade Educator

Problem 89

If $p$ and $q$ are chosen randomly from the set $\{1,2,3,4,5,6,7,8,9$ $10\}$, with replacement, determine the probability that the roots of the equation $x^{2}+p x+q=0$ are real.

Aayush Gupta

Numerade Educator

Problem 90

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?

Aman Gupta

Numerade Educator

Problem 91

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 $2,3,4, \ldots . .12$ is picked and the number on the card is noted. What is the probability that the noted number is either 7 or $8 ?$

Aman Gupta

Numerade Educator

Problem 92

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
[1992-6 Marks] $A=($ the first bulb is defective $)$ $B=($ the second bulb 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

Aayush Gupta

Numerade Educator

Problem 93

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 question given that he correctly answered it.

Aayush Gupta

Numerade Educator

Problem 94

$\mathrm{A}$ 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.

Aayush Gupta

Numerade Educator

Problem 95

A box contains 2 fifty paise coins, 5 twenty five paise coins and a certain fixed number $N(\geq 2)$ 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.

Aayush Gupta

Numerade Educator

Problem 96

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.

Aayush Gupta

Numerade Educator

Problem 97

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.

Aayush Gupta

Numerade Educator

Problem 98

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?

Aayush Gupta

Numerade Educator

Problem 99

$A, B, C$ are events such that
$P(A)=0.3, P(B)=0.4, P(C)=0.8$
$P(A B)=0.08, P(A C)=0.28 ; \quad P(A B C)=0.09$If $P(A \cup B \cup C) \geq 0.75$, then show that $P(B C)$ lies in the interval $0.23 \leq x \leq 0.48$

Aman Gupta

Numerade Educator

Problem 100

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 $P_{r}\{N=n\}=\frac{(n-1)(52-n)(51-n)}{50 \times 49 \times 17 \times 13}$ where $2 \leq n \leq 50$

Aman Gupta

Numerade Educator

Problem 101

$A$ 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 $0.9$ ?

Aman Gupta

Numerade Educator

Problem 102

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 ?[ 1981
- 2 Marks]

Aman Gupta

Numerade Educator

Problem 103

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.

Aman Gupta

Numerade Educator