Comprehensive Mathematics for JEE Advanced

Ravi Prakash; Ajay Kumar; Usha Gupta

Chapter 9

Probability - all with Video Answers

Educators

Chapter Questions

Problem 1

A group consists $2 n$ students including you and your fricnd. If the group is split into two different sections $A$ and $B$, each containing $n$ students, then the probability that you and your friend are in the different sections is
(a) $\frac{n}{2 n-1}$
(b) $\frac{n-1}{2 n-1}$
(c) $\frac{n+1}{2 n-1}$
(d) $\frac{2}{2 n-1}$

Aman Gupta

Aman Gupta

Numerade Educator

Problem 2

Two persons $A$ and $B$ think of two numbers at random from the numbers $1,2, \ldots, n$. Probability $A$ think of a number smaller than thought by $B$ is
(a) $\frac{n-1}{2 n}$
(b) $\frac{2 n-1}{2 n}$
(c) $\frac{n-1}{2 n}$
(d) $\frac{n}{2 n-1}$

Aman Gupta

Numerade Educator

Problem 3

A box contains 20 cards of these 10 have letter $J$ printed on them and the remaining 10 have $E$ print-ed on them. 3 cards are drawn from the box, the probability that we can write $J E E$ with these cards is
(a) $9 / 80$
(b) $1 / 8$
(c) 427
(d) $15 / 38$

Aman Gupta

Numerade Educator

Problem 4

$A$ and $B$ toss a coin alternatively till one of them gets a head and wins the game. If $A$ begins the game, the probability $B$ wins the game is
(a) $1 / 2$
(b) $1 / 3$
(c) $1 / 4$
(d) $2 / 3$

Aman Gupta

Numerade Educator

Problem 5

A fair coin is tossed repeatedly. If head and tail appear alternatively on first 7 tosses; then the probability that head appears on the eighth toss is
(a) $1 / 2$
(b) $1 / 128$
(c) $1 / 256$
(d) $7 / 256$

Aman Gupta

Numerade Educator

Problem 6

If $P(A)>0$, then the event $A$ is independent of itself if and only if $P(A)$ is
(a) $1 / 3$
(b) $1 / 2$
(c) 1
(d) $(\sqrt{5}-1) / 2$

Aman Gupta

Numerade Educator

Problem 7

Two numbers are selected at random from $1,2,3, \ldots .$ 100 and are multiplied, then the probability (correct to two places of decimals) that the product thus obtnined is divisible by 3 is
(a) $0.22$
(b) $0.33$
(c) $0.44$
(d) $0.55$

Aman Gupta

Numerade Educator

Problem 8

The probability that an event $A$ occurrs in a single trial of an experiment is $0.6$. Three independent trials of the experiment are performed. The probability that the event $A$ occurs at least twice is
(a) $0.636$
(b) $0.632$
(c) $0.648$
(d) $0.946$

Aman Gupta

Numerade Educator

Problem 9

Suppose $X-B(n, p)$ and $P(X=3)=P(X=5) .$ If $p>$ $1 / 2$, then
(a) $n \leq 7$
(b) $n>8$
(c) $n \geq 9$
(d) $n \geq 10$

Aman Gupta

Numerade Educator

Problem 10

In three throw of a pair dice, the probability throwing doublets not more than twice is
(a) $1.6$
(b) $5 / 72$
(c) $215 / 216$
(d) $7 / 128$

Aman Gupta

Numerade Educator

Problem 11

A bag contains 4 brown and 5 white socks, $A$ man pulls two socks at random without replacement. The probability that the man gets both the socks of the same colour is
(a) $5 / 108$
(b) $1 / 6$
(c) $5 / 18$
(d) $4 / 9$

Aman Gupta

Numerade Educator

Problem 12

$A$ is a set containing $n$ elements. Two subsets $P$ and $Q$ of $A$ are chosen at random. $(P$ and $Q$ may have elements in common). The probability that $P \cup Q \neq A$ is
(a) $(3 / 4)^{\prime}$
(b) $1 / 4 n$
(c) ${ }^{n} C_{2} / 2^{n}$
(d) $1-(3 / 4)^{\text {m }}$

Aayush Gupta

Numerade Educator

Problem 13

Three of the six vertices of a regular hexagon are chosen at random. The probability that the triangle with these vertices is not equilateral is
(a) $1 / 2$
(b) $1 / 5$
(c) $9 / 10$
(d) $1 / 20$

Aman Gupta

Numerade Educator

Problem 14

A positive integer is chosen at random. The probability that the sum of the digits of its square is 39 is
(a) $1 / 39$
(b) $2 / 39$
(c) $1 / 11$
(d) 0

Aman Gupta

Numerade Educator

Problem 15

A lottery sells $n^{2}$ tickets and declares $n$ prizes. If a man purchases $n$ tickets, the probability of his winning at least one prize is
(a) $\left(n^{2}-n\right) ! /\left(n^{2}\right) !$
(b) $1 / 2 n$
(c) $(n-1) t^{2} /\left(n^{2}\right) !$
(d) none of these

Aman Gupta

Numerade Educator

Problem 16

Let $x$ be a non-zero real number. A determinant is chosen from the set of all deteminants of order 2 with entries $x$ or $-x$ only. The probability that the value of the determinant is non-zero is
(a) $3 / 16$
(b) $1 / 4$
(c) $1 / 2$
(d) $1 / 8$

Anas Venkitta

Numerade Educator

Problem 17

A die is rolled three times. The probability of getting a number larger than the previous number ench time is
(a) $5 / 72$
(b) $5 / 54$
(c) $13 / 216$
(d) $1 / 18$

Aman Gupta

Numerade Educator

Problem 18

In a game called "odd man out man out," $m(m>2)$ persons toss a coin to determiane who will buy refreshments for the entire group. A person who gets an outcome different from that of the rest of the members of the group is called the odd man out. The probability that there is a loser in any game is
(a) $1 / 2 m$
(b) $m / 2^{n-1}$
(c) $2 / m$
(d) $1 / 2^{m-1}$

Aman Gupta

Numerade Educator

Problem 19

Two non-negative integers are chosen at random. The probability that the sum of the square is divisible by 11 is
(a) $9 / 16$
(b) $1 / 121$
(c) $9 / 17$
(d) $2 / 121$

Aman Gupta

Numerade Educator

Problem 20

A natural number $x$ is choosen at random from the first 100 natural numbers. The probability that
$$
x+\frac{100}{x}>50
$$
is
(a) $1 / 10$
(b) $11 / 50$
(c) $11 / 20$
(d) $3 / 20$

Aman Gupta

Numerade Educator

Problem 21

A bag contains $(2 n+1)$ coins. It is known that $n$ of these coins have a hend on both sides, whereas the remaining $(n+1)$ coins are fair. $A$ coin is picked up at random from the bag and tossed. If theprobubility that the toss results in a head is $31 / 42$, then $n$ is equal to
(a) 10
(b) 11
(c) 12
(d) 13

Aman Gupta

Numerade Educator

Problem 22

If $X$ and $Y$ are independent binomial variate $B(5,1 / 2)$ and $B(7,1 / 2)$, then $P(X+Y=3)$ is
(a) $55 / 1024$
(b) $55 / 4098$
(c) $55 / 2048$
(d) $55 / 512$

Aman Gupta

Numerade Educator

Problem 23

Suppose $n(\geq 3)$ persons are sitting in a row. Two of them are sclected at random. The probability that they are not together is
(a) $1-\frac{2}{n}$
(b) $\frac{2}{n-1}$
(c) $1-\frac{1}{n}$
(d) $\frac{2}{n}$

Aman Gupta

Numerade Educator

Problem 24

A bag contains four tickets marked with numbers 112, 121, 211, 222. One ticket is drawn at random from the bag. Let $E_{i}(i=1,2,3)$ denote the event that $i$ th digit on the ticket is 2 . Then which of the following is not true
(a) $E_{1}$ and $E_{2}$ are independent
(b) $E_{2}$ and $E_{3}$ are independent
(c) $E_{3}$ and $E_{\mathrm{j}}$ are independent
(d) $E_{\mathrm{j}}, E_{2}, E_{3}$ are independent

Aman Gupta

Numerade Educator

Problem 25

If the letters of the word PROBABILITY are written down at random in a row, the probability that two $\mathrm{B}^{\prime} \mathrm{s}$ are together is
(a) $2 / 11$
(b) $10 / 11$
(c) 311
(d) $6 / 11$

Aman Gupta

Numerade Educator

Problem 26

If four positive integers are taken at random and multiplicd together, then the probability that the last digit is $1,3.7$ or 9 is
(a) $1 / 8$
(b) $2 / 7$
(c) $1 / 625$
(d) $16.625$

Aman Gupta

Numerade Educator

Problem 27

Two contestants play a game as follows: each is asked to select a digit from 1 to 9 . If the two digits match they both win a prize. The probability that they will win a prize in a single trial is
(a) $1 / 81$
(b) $7 / 81$
(c) $1 / 9$
(d) $3 / 11$

Aman Gupta

Numerade Educator

Problem 28

A fair coin is tossed $2 n$ times. Probubility of getting more heads than tails is
(a) $1 / 2$
(b) ${ }^{2 n} C_{n}\left(\frac{1}{2}\right)^{2 n}$
(c) $1-2^{\prime \prime} C_{n}\left(\frac{1}{2}\right)^{2 n}$
(d) $\frac{1}{2}-{ }^{2 n} C_{n}\left(\frac{1}{2}\right)^{2 n+1}$

Aman Gupta

Numerade Educator

Problem 29

If $A$ and $B$ are two events, then which one of the following is not always true
(a) $P(A \cap \mathrm{B}) \leq P(A)+P(B)-1$

Aman Gupta

Numerade Educator

Problem 30

A sum of money is rounded off to the nearest rupee. The probability that the round off error is at most 10 paise is
(a) $1 / 10$
(b) $11 / 100$
(c) $3 / 25$
(d) $21 / 100$

Aman Gupta

Numerade Educator

Problem 31

The probability that an electric bulb will last 150 days or more is $0.7$ and that it will last at most 160 days is $0.8$. The probability that the bulb will last 150 to 160 days is
(a) $0.5$
(b) $1 / 3$
(c) $0.56$
(d) $0.59$

Aman Gupta

Numerade Educator

Problem 32

A bag contains three tickets numbered 1,2 and $3 .$ A ticket is drawn at random and put back in the bag, and this is done four times, The probability that the sum of the numbers drawn is even is
(a) $40 / 81$
(b) $41 / 81$
(c) $14 / 27$
(d) $13 / 81$

Aman Gupta

Numerade Educator

Problem 33

Two numbers $x$ and $y$ are selected at random from the set $\{1,2,3, \ldots, 3 N\}$. The probability that $x^{2}-y^{2}$ is divisible by 3 is
(a) $(3 N-1) / 3 N$
(b) $(N-1) / N$
(c) $(5 N-3) /(9 N-3)$
(d) $(N-1) / 2 N$

Aman Gupta

Numerade Educator

Problem 34

Three persons $A, B$ and $C$ are to speak at a function along with 7 other persons. The probability that $A$ speaks before $B$ and $B$ speaks before $C$ is
(a) $3 / 70$
(b) $1 / 6$
(c) $3 / 7$
(d) $1 / 9$

Aman Gupta

Numerade Educator

Problem 35

$n$ men and $n$ women are seated at round table in random order. The probalility that they can be divided into $n$ non-interrecting pairs so that each pair consists of a man and a woman is
(a) $1 / 2 n$
(b) $2\left(2^{n}-1\right)^{2 n} c_{a}$
(c) $2 n^{2 n} C_{n}$
(d) $1 /\left({ }^{\prime \prime} C_{n}\right)^{2}$

Aman Gupta

Numerade Educator

Problem 36

If $a$ is an integer lying in $[-5,30]$. then the probability that the graph of $y=x^{2}+2(a+4) x$ $-5 a+64$ is strictly above the $x-a x$ is is
(a) $1 / 6$
(b) $7 / 36$
(c) $2 / 9$
(d) $3 / 5$.

Aman Gupta

Numerade Educator

Problem 37

Two numbers $X$ and $Y$ are chosen at random (without replacement) from the set $\{1,2, \cdots 5 N\}$. The probability that $X^{4}-Y^{4}$ is divisible by 5 is
(a) $\frac{N-1}{5 N-1}$
(b) $\frac{4(4 N-1)}{5(5 N-1)}$
(c) $\frac{17 N-5}{5(5 N-1)}$
(d) $\frac{3 N-5}{5(5 N-1)}$.

Aman Gupta

Numerade Educator

Problem 38

There is a five-volume dictionary among 50 books arranged on a shelf in random order. The probability that these volumes stand in increasing order from left to right (the volumes are not necessarily kept side-by-side) is
(a) $1 / 5$
(b) $1 / 5^{\text {so }}$
(c) $1 / 50^{5}$
(d) $1 / 120$.

Aman Gupta

Numerade Educator

Problem 39

A speaks the truth in 70 percent cases and $B$ in 80 percent eases. The probability that they will contradict cach other in describing a single event is
(a) $0.36$
(b) $0.38$
(c) $0.4$
(d) $0.42$.

Aman Gupta

Numerade Educator

Problem 40

$2 n$ boys are randomly divided into two subgroups containing $n$ boys each. The probability that the two tallest boys are in different groups is
(a) $\frac{n}{2 n-1}$
(b) $\frac{n-1}{2 n-1}$
(c) $\frac{2 n-1}{4 n^{2}}$
(d) $\frac{n-1}{2 n^{2}}$.

Aman Gupta

Numerade Educator

Problem 41

The numbers $1,2,3, \cdots, n$ are arranged in random order. The probability that the digits $1,2, \cdots, k(k<$ $n$ ) appears as neighbours in that order is
(a) $\frac{1}{n !}$
(b) $\frac{k !}{n !}$
(c) $\frac{(n-k) !}{n !}$
(d) $\frac{(n-k+1) !}{n !}$

Aman Gupta

Numerade Educator

Problem 42

A random variable $X$ has the probability distribution:
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline$x$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline$P(X-x)$ & $0.15$ & $0.23$ & $0.12$ & $0.10$ & $0.20$ & $0.08$ & $0.07$ & $0.05$ \\
\hline
\end{tabular} For the events $E=\{x \mid x$ is prime\} and $F=\langle x, x<4\}$, the probability $P(E \cup F)$ is
(a) $0.35$
(b) $0.77$
(c) $0.87$
(d) $0.50$

Aman Gupta

Numerade Educator

Problem 43

Two boys and three girls stand in a queue. The probability the number of boys ahead of every girl is at least one more than the number of girls ahead of her
(a) $1 / 2$
(b) $1 / 3$
(c) $2 / 3$
(d) 0

Aman Gupta

Numerade Educator

Problem 44

A bag contains some white and some black balls, all combinations of balls being equally likely. The total number of balls in the bag is 10 . If three balls are drawn at random without replacement and all of them are found to be black, the probability that the bag contains 1 white and 9 black balls is
(a) $14 / 55$
(b) $12 / 55$
(c) 2111
(d) $8 / 55$

Anas Venkitta

Numerade Educator

Problem 45

Let $E, F, G$ be pairwise independent events with $P(G)>0$ and $P(E \cap F \cap G)=0$Then $P\left(E^{*} \cap F^{4} \mid G\right)$ equals
(a) $P\left(E^{\prime}\right)+P\left(F^{\prime}\right)$
(b) $P\left(E^{\prime}\right)-P\left(F^{7}\right)$
(c) $P\left(E^{*}\right)-P(F)$
(d) $P(E)-P\left(F^{\prime}\right)$

Aman Gupta

Numerade Educator

Problem 46

$A$ and $B$ are two events, The probability that at most one of $A, B$ occurs is
(a) $1-P(A \cap B)$
(b) $P\left(A^{\prime}\right)+P\left(B^{\prime}\right)-P\left(A^{\prime} \cap B^{\prime}\right)$
(c) $P\left(A^{\prime}\right)+P\left(B^{\prime}\right)+P(A \cup B)-1$
(d) $P\left(A \cap B^{\prime}\right)+P\left(A^{\prime} \cap B\right)+P\left(A^{*} \cap B^{\prime}\right)$.

Aman Gupta

Numerade Educator

Problem 47

The probability of the simultancous oecurrence of two events $A$ and $B$ is $p$. If the probability that exactly one of $A$, $B$ occurs is $q$, then
(a) $P\left(A^{\prime}\right)+P\left(B^{\prime}\right)=2+2 q-p$
(b) $P\left(A^{\prime}\right)+P\left(B^{\prime}\right)=2-2 p-q$
(c) $P(A \cap B \mid A \cup B)=\frac{p}{p+q}$
(d) $P\left(A^{\prime} \cap B^{\prime}\right)=1-p-q$.

Aman Gupta

Numerade Educator

Problem 48

Suppose that $P(A)=3 / 5$ and $P(B)=2 / 3$. Then
(a) $P(A \cup B) \geq 2 / 3$
(b) $4 / 15 \leq P(A \cap B) \leq 3 / 5$
(c) $2 / 5 \leq P(A \mid B) \leq 9 / 10$
(d) $P\left(A \cap B^{\prime}\right) \leq 1 / 3$,

Aman Gupta

Numerade Educator

Problem 49

If $E$ and $F$ are independent events such that $0<P(E)<1$ and $0<P(F)<1$, then
(a) $E$ and $F$ are mutually exclusive
(b) $E$ and $F^{\circ}$ are independent
(c) $E^{*}$ and $F^{*}$ are independent
(d) $P(E \mid F)+P\left(E^{*} \mid F\right)=1$.

Aman Gupta

Numerade Educator

Problem 50

If $A$ and $B$ are two events, then
(a) $P(A \cap B) \leq \min \{P(A), P(B)\}$
(b) $P(A \cap B) \cap \max \left\{0,1-P\left(A^{\prime}\right)-P\left(B^{\prime}\right)\right\}$
(c) $P(A \cap B) \leq P(A \cup B)$
(d) $P(A \cap B)=P(A) P(B)$ if $A$ and $B$ are independent.

Aman Gupta

Numerade Educator

Problem 51

Twenty children are standing in a line outside a ticket window at Appu Ghar in New Delhi. Ten of these children have a one-rupee coin each and the remaining 10 have a two rupee coin each. The entry ticket is priced at Re. 1. If all the arrange-ments of the 20 children are equally likely, the probability that

Sanchit Jain

Numerade Educator

Problem 52

Eight tickets numbered $000,010,011,011,100$, 101, 101 and 110 are placed in a bag. One ticket is drawn from the bag at random. Let $A, B$ and $C$ denote the following events: $A$ - "the first digit is $0^{* *} B$ "the second digit is $0^{* *}$ and $C-$ "the third digit is $0^{+4}$. Then
(a) $A$ and $B$ are independent
(b) $B$ and $C$ are independent
(c) $C$ and $A$ are indcpendent
(d) $A, B, C$ are independent

Aman Gupta

Numerade Educator

Problem 53

Four natural numbers are taken at random and are multiplied. Let $p_{1}$ denote the probability of the event that last digit of the product is $1,3,7$ or 9 , and $p_{2}$ denote the probability that the lnst digit of the product is 5 , then
(a) $p_{y}>p_{2}$
(b) $p_{1}=p_{1}$
(c) $p_{i}<p_{2}$
(d) $p_{1}+p_{2}<1 / 2$

Aman Gupta

Numerade Educator

Problem 54

I wrote a letier to my fricnd $X$ and gave it to my son to post it. The probability that my son will forget to post the letter is $1 / 10$ and the letter will be lost in the post is $1 / 100 .$ If my friend $X$ did not receive the letter, then the probability that
(a) my son forgot to post the letter is $891 / 1000$
(b) the letter was lost in the post $891 / 991$
(c) my son forgot to post the letter is $100 / 991$
(d) none of these.

Aman Gupta

Numerade Educator

Problem 55

A number $x$ is chosen at random from the first 100 natural numbers. The probability that it satisfies
(a) $x^{2}-25 x \leq 150$ is $0.3$
(b) $x^{2}-17 x+30 \geq 0$ is $0.88$
(c) $30 x-x^{2}$ is a perfeet square of a natural number is $0.07$
if) $20 \mathrm{r}$
$4 .$

Aman Gupta

Numerade Educator

Problem 56

Two fair dice are thrown. The probability that difference between the number is
(a) two is $2 / 9$
(b) three is $1 / 6$
(c) at least two is $1 / 2$
(d) at most one is $4 / 9$

Aman Gupta

Numerade Educator

Problem 57

I post a letter to my friend and do not receive a reply. It is known that one letter out of $m$ letters do not reach its destination. If it is certain that my friend will reply if he receives the letter. If $A$ denotes the event that $\mathrm{my}$ friend receives the letter and $B$ that 1 get a reply, then

Victoria Dollar

Victoria Dollar

Numerade Educator

Problem 58

An electric component manufactured by 'KIAANElectronics' is tested for its defectiveness by a sophisticated testing device. Let $A$ denote the event "the device is defective" and $B$ the event "the testing device reveals the component to be defective". Suppose $P(A)=\alpha$ and $P(B \mid A)=P\left(B^{\prime} \mid A^{\prime}\right)=1-\alpha$,
where $0<\alpha<1$, then
(a) $P(B)=2 \alpha(I-\alpha)$
(b) $P\left(A^{\prime} \mid B\right)=1 / 2$
(c) $P\left(B^{\prime}\right)=(1-\alpha)^{2}$
(d) $P\left(A^{\prime} \mid B^{\prime}\right)=[\alpha(1-\alpha)]^{2}$

Hunza Gilgit

Numerade Educator

Problem 59

Suppose $X$ is a random variable which takes values 0, $1,2,3, \ldots$ and $P(X=r)=p q^{r}$ where $0<p<1, q$
$=1-p$ and $r=0,1,2, \ldots .$ Then
(a) $P(X \geq n)=q^{\prime \prime}$
(b) $P(X \geq m+n \mid X \geq m)=P(X \geq n)$
(c) $P(X=m+n \mid X \geq m)=P(X=n)$
(d) $P(X>n)=q^{n+1}$

Aman Gupta

Numerade Educator

Problem 60

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

Aman Gupta

Numerade Educator

Problem 61

A die is rolled three times. Let $E_{1}$ denote the event of getting a number larger than the previous number each time and $E_{y}$ denote the event that the numbers (in order) from an increasing A.P. then
(a) $P\left(E_{2}\right) \leq P\left(E_{1}\right)$
(b) $P\left(E_{2} \cap E_{1}\right)=1 / 36$
(c) $P\left(E_{2}\left[E_{1}\right)=\frac{3}{10}\right.$
(d) $P\left(E_{1}\right)=\frac{10}{3} P\left(E_{2}\right)$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 62

Seven digits from the numbers $1,2,3, \ldots, 9$ are written in random order to obtain a seven digit number
N. The probability that $N$ is divisible by
(a) 9 is $41^{9} P_{7}$
(b) 3 is $35 \mathrm{~F} P_{7}$
(c) 5 is $1 / 3$
(d) 5 is $1 / 4$

Anas Venkitta

Numerade Educator

Problem 63

Eight children are standing in a queue outside a ticket window at Appu Ghar in New Delhi. Four of these children have a one-rupee coin cach and the remaining 4 children have a two-rupee coin each. The entry ticket is priced at Re. $1 .$ If all the arrangements of the cight children are equally likely, let $p_{1}$ be probability that the ith child will be the first to wait for change, if the cashier at the ticket window has no change to begin with, then(a) $p_{1}=1 / 2$
(b) $p_{2}=0$
(c) $p_{1}=1 / 7$
(d) probability that no child has to wait for the change is $1 / 5$

Sherrie Fenner

Numerade Educator

Problem 64

If $X$ follows a binomial distribution with parameter $n=101$ and $p=1 / 3$, then $P(X=r)$ is maximum if $r$ cquals
(a) 34
(b) 33
(c) 32
(d) 35

Aman Gupta

Numerade Educator

Problem 65

When a fair die is thrown twice, let $(a, b)$ denote the outcome in which the first throw shows $a$ and the second shows $b$. Further, let $A, B$ and $C$ be the following events: $A=I(a, b) \mid a$ is odd $\}$. $B=\{(a, b) \mid b$ is odd $\}$ and $C=\{(a, b) \mid a+b$ is odd $\}$. Then
(a) $P(A \cap B)=1 / 4$
(b) $P(B \cap C)=1 / 4$
(c) $P(A \cap C)=1 / 4$
(d) $P(A \cap B \cap C)=0$

Aman Gupta

Numerade Educator

Problem 66

Three players $A, B$ and $C$ alternatively throw a die in that order, the first player to throw a 6 being deemed the winner, $\mathrm{A}$ 's die is fair whereas $B$ and $\mathrm{C}$ throw dice with probabilities $p_{1}$ and $p_{2}$ respectively of throwing a 6 . Column 1 Column 2
(a) If $p_{1}=1 / 5, p_{2}=1 / 4$,
(p) $\frac{1}{2}$
the probubility that $A$ wins the game.
(b) If $p_{1}=1 / 5, p_{2}=1 / 4$
(q) $\frac{1}{3}$
the probability that $C$ wins the game
(c) Value of $p_{1}$ so that $P(A$ wins $)=P(B$ wins $)$
(r) $\frac{1}{4}$
(d) Value of $p_{2}$ so that the game is equiprobable to
(s) $\frac{1}{5}$
all the three players.

Aakash Goyal

Numerade Educator

Problem 67

Consider a town with $N$ people. A person spreads a rumor to a second, who in turn repeats it to a third. and so on. Suppose that at each stage, the recipient of the rumor is chosen at random from the remaining $(N-1)$ people. Suppose rumor is repeated $n(\geq 3)$ times, then probability that it will not be repeated to(a) originator
(p) $\left(1-\frac{1}{N-1}\right)$
(b) first recipicnt
(q) $\left(1-\frac{1}{N-1}\right)^{n-2}$
(c) second recipient
(r) $\left(1-\frac{1}{N+1}\right)^{n-2}$
(d) any of the carlier
(s) ${ }^{N-1} P_{n} /(N-1)^{n}$
recipient

Amy Jiang

Numerade Educator

Problem 68

A prisoner escapes from a jail and is equally likely to choose one of the four roads $1, \mathrm{II}$, III or IV to reach away from the hands of law. If he choose 1 road, he is successful with probability $1 / 6$ and for II, III and IV this is $1 / 8,1 / 10$ and $1 / 12$. If the prisoner is successful, the probability that he chose road Column 1 $\quad$ Column 2
(a) I
(p) $12 / 57$
(b) II
(q) $15 / 57$
(c) III
(r) $20 / 57$
(d) IV
(s) $10 / 57$

MB

Matt Bremer

Numerade Educator

Problem 69

Sixteen players $S_{1}, S_{2}, \ldots ., S_{16}$ play in a tournament. They are divided into eight pairs at random. From each pair a winner is decided on the basis of a game played between the two players. Assume that all the players are of equal strength. The probability that Column 1 Column 2
(a) $S_{1}$ reaches the second round.
(p) $\frac{1}{2}$
(b) $S_{2}$ reaches the second round
(q) $\frac{1}{3}$
(c) Exactly one of $S_{1}$ and $S_{2}$ reach the
(r) $\frac{8}{5}$
second round.
(d) Both $S_{1}$ and $S_{2}$ are among the winners
(s) $\frac{7}{30}$

Aman Gupta

Numerade Educator

Problem 70

Three distinct numbers $a, b, c$ are chosen at random from the numbers $1,2, \ldots \ldots ., 100$. The probability that Column 1 $\quad$ Column 2
(a) $a, b, c$ are in A.P.
(p) $\frac{53}{161700}$(b) $a, b, c$ are in G.P.
(q) $\frac{1}{66}$
(c) $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in G.P.
(r) $\frac{1}{22}$
(d) $a+b+c$ is
(s) $\frac{1}{2}$
divisible by 2

Aman Gupta

Numerade Educator

Problem 71

Let $A, B$ and $C$ be three events such that $P(C)=0$. Statement-1: $P(A \cap B \cap C)=0$
Statement-2: $P(A \cup B \cup C)=P(A \cup B)$

Aman Gupta

Numerade Educator

Problem 72

Let $A, B$ and $C$ be three events such that $A$ is independent of both $B$ and $C$. Statement-1: $A$ is independent of $B \cup C$ Statemtne- $2: A$ is independent of $B \cap C$

Aman Gupta

Numerade Educator

Problem 73

$\mathrm{A}$ set $P$ contains $n$ elements. Two boys Rakshit and Kiaan independently pick up two subsets $Q$ and $R$ of $P$ Statement-1: Probability $Q \cap R=\phi$ is $\left(\frac{3}{4}\right)^{n}$
Statement-2: Probability $Q \cup R=P$ is $\left(\frac{1}{2}\right)^{n}$.v

Aman Gupta

Numerade Educator

Problem 74

A fair die is thrown twice, Let $(a, b)$ denote the outcome in which the first throw shows $a$ and the second shows $b$. Let $A$ and $B$ be the following two cvents. $A=\{(a, b) \mid a$ is even $\}, B=[(a, b) \mid b$ is even $\}$ Statement-1: If $C=\{(a, b) \mid a+b$ is odd $\}$, then $P(A \cap B \cap C)=1 / 8$
Statement-2: If $D=\{(a, b) \mid a+b$ is even $\}$, then $P(A \cap B$ $\cap D \mid A \cap B)=1 / 3$

Aman Gupta

Numerade Educator

Problem 75

A $\operatorname{man} P$ speaks truth with probability $p$ and another $\operatorname{man} Q$ speaks truth with probability $2 p$. Statement-1: If $P$ and $Q$ contradict each other with probability $1 / 2$, then there are two values of $p$. Statement-2: A quadratic equation with real coefficients has two real roots.

Aman Gupta

Numerade Educator

Problem 76

$p(0)$ equals
(a) $w \bar{b}$
(b) $w(b+w)$
(c) $b / w$
(d) 0

Anas Venkitta

Numerade Educator

Problem 77

$p(1)$ equals
(a) $\mathrm{w} / \mathrm{b}$
(b) $(w-1)(b+w-1)$
(c) $w(b+w)$
(d) 0

Muhammad Nawaz

Numerade Educator

Problem 78

$p(2)$ equals
(a) $w(b+w)$
(b) $(w-2) /(b+w-2)$
(c) $w(w-2)(b+w)(b+w-1)$
(d) 0

Raj Bala

Numerade Educator

Problem 79

If $l>2, p(l)$ equals
(a) $w /(b+w)$
(b) $(w-1) / b+w-1)$
(c) $\frac{" P_{f}}{m+b_{f}}$
(d) none of these

Wendi Zhao

Numerade Educator

Problem 80

If $b$ is fixed then $\lim _{\infty \rightarrow \infty} p(l)$ equals
(a) 1
(b) 0
(c) $1 / 2$
(d) $l / \mathrm{w}$

Abhijith V

Numerade Educator

Problem 81

If $r=0, p_{n}$ equals
(a) $2 \mathrm{k} / \mathrm{n}$
(b) $(k+1) / n$
(c) $(2 k+1) / n$
(d) $\mathrm{k} / \boldsymbol{n}$

Aman Gupta

Numerade Educator

Problem 82

If $r=9, p_{n}$ equals
(a) $2 k i n$
(b) $2(k+1) n$
(c) $(2 k+1) / n$
(d) $k / n$

Muhammad Saleem

Muhammad Saleem

Numerade Educator

Problem 83

If $1 \leq r \leq 8, p_{n}$ equals
(a) $(2 k-1) / n$
(b) $2 k$ in
(c) $(2 k+1) / n$
(d) $\mathrm{k} / \boldsymbol{n}$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 84

$\lim _{n \rightarrow-} p_{n}$ equals
(a) $1 / 10$
(b) $2 / 5$
(c) $1 / 5$
(d) $3 / 5$

Gaurav Kalra

Numerade Educator

Problem 85

If $q_{n}$ denote the probability that $a^{2}+1$ is divisible by 10, then $\lim _{n \rightarrow+\infty} q_{n}$ equals
(a) $1 / 5$
(b) $2 / 5$
(c) $3 / 5$
(d) 0

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 86

$p_{1}$ cquals
(a) $1 / 6$
(b) 1
(c) 0
(d) $1 / 5$

Monica Miller

Numerade Educator

Problem 87

$p_{2}$ cqual
(a) $1 / 6$
(b) 1
(c) 0
(d) $1 / 5$

Monica Miller

Numerade Educator

Problem 88

If $2 \leq r \leq 6, p_{y}$, equals
(a) $1-\left(\frac{1}{6}\right)^{r-1}$
(b) $\frac{1}{5}\left[1-\left(\frac{1}{6}\right)^{r-1}\right]$
(c) $5^{r} / 6^{r}$
(d) $1 / 5$

Gaurav Kalra

Numerade Educator

Problem 89

If $r>6, p_{x}$ equals
(a) $1-\left(\frac{1}{6}\right)^{r-1}$
(b) $\frac{1}{5}\left[1-\left(\frac{1}{6}\right)^{r-1}\right]$
(c) $\frac{1}{5}\left[\left(\frac{1}{6}\right)^{r-6}-\left(\frac{1}{6}\right)^{r-1}\right]$
(d) $1 / 5$

Gaurav Kalra

Numerade Educator

Problem 90

Sum of the series $S=\sum_{r=1} p_{r}$ is
(a) 1
(b) $1 / 6$
(c) $1 / 5$
(d) $2 / 3$

Aman Gupta

Numerade Educator

Problem 91

L.ct $P(A)=0.4$ and $P(A \cup B)=P(A \cap B)$, then $5 P(B)$ is

Aman Gupta

Numerade Educator

Problem 92

A pair of fair dice is rolled together till a sum of 7 or 11 is obtained. Let $p$ denote the probability that 7 comes before 11 , then the value of $4 p$ is

Aman Gupta

Numerade Educator

Problem 93

In a test an examine cither gusses or copies or knows the answer to a multiple choice question with $m$ choices out of which cxactly one is correct. The probability that he makes 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 $1 / 8$. If the probability that he knew the answer to the question given that he correctly answered it is $120 / 141$, then $m$ is

Aman Gupta

Numerade Educator

Problem 94

If $\frac{1}{4}(1+4 p), \frac{1}{4}(1-p)$ and $\frac{1}{2}(1-2 p)$ are the
probabilities of three mutually exclusive events, then $2 p$ is

Aman Gupta

Numerade Educator

Problem 95

The digits $1,2,3,4,5,6,7,8$ and 9 are written in random order to form a nine digit number. Let $p$ be the probability that this number is divisible by 36 , then $9 p$ is

Goutam Chand

Numerade Educator

Problem 96

Two squares are chosen at random on a chessboard. If $p$ denotes the probability that they have exactly one side in common, then $144 p$ is

Aman Gupta

Numerade Educator

Problem 97

In a multiple-choice question, there are five alternative answers, of which one or more than one are correct. A candidate will get marks on the question if he ticks all the correct answers. If he decides to tick answers at random, then the least number of choices should he be allowed so that the probability of his getting marks on the question exeeeds $1 / 8$ is

Aman Gupta

Numerade Educator

Problem 98

Two persons ench make a single throw with a pair of dice. If $p$ denotes the probability that their throws are unequal, then $648 p-70$ is

Aman Gupta

Numerade Educator

Problem 99

In a sequence of independent trials, the probability of success is $1 / 4$. If $p$ denotes the probability that the second success occurs on the fourth trial or later trial, then $32 p-20$ is

Aman Gupta

Numerade Educator

Problem 100

If $X$ follows a binomial distribution with parameters $n=100$ and $p=1 / 3$, then $r-30$ for which $P(X-r)$ is maximum is

Aman Gupta

Numerade Educator