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 $B$ is green, then the probability that the original signal was green is
A) $\frac { 3 } { 5 }$
B) $\frac { 6 } { 7 }$
C) $\frac { 20 } { 23 }$
D) $\frac { 9 } { 20 }$

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 $\omega ^ { r _ { 1 } } + \omega ^ { r _ { 2 } } + \omega ^ { r _ { 3 } } = 0$ is
A) $\frac { 1 } { 18 }$
B) $\frac { 1 } { 9 }$
C) $\frac { 2 } { 9 }$
D) $\frac { 1 } { 36 }$

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) $\frac { 116 } { 181 }$
(B) $\frac { 126 } { 181 }$
(C) $\frac { 65 } { 181 }$
(D) $\frac { 55 } { 181 }$

Let $X$ and $Y$ be two events such that $P(X) = \frac{1}{3}$, $P(X \mid Y) = \frac{1}{2}$ and $P(Y \mid X) = \frac{2}{5}$. Then
[A] $P(Y) = \frac{4}{15}$
[B] $P(X' \mid Y) = \frac{1}{2}$
[C] $P(X \cap Y) = \frac{1}{5}$
[D] $P(X \cup Y) = \frac{2}{5}$

There are three bags $B _ { 1 } , B _ { 2 }$ and $B _ { 3 }$. The bag $B _ { 1 }$ contains 5 red and 5 green balls, $B _ { 2 }$ contains 3 red and 5 green balls, and $B _ { 3 }$ contains 5 red and 3 green balls. Bags $B _ { 1 } , B _ { 2 }$ and $B _ { 3 }$ 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?
(A) Probability that the chosen ball is green, given that the selected bag is $B _ { 3 }$, equals $\frac { 3 } { 8 }$
(B) Probability that the chosen ball is green equals $\frac { 39 } { 80 }$
(C) Probability that the selected bag is $B _ { 3 }$, given that the chosen ball is green, equals $\frac { 5 } { 13 }$
(D) Probability that the selected bag is $B _ { 3 }$ and the chosen ball is green equals $\frac { 3 } { 10 }$

Consider three sets $E_1 = \{1,2,3\}$, $F_1 = \{1,3,4\}$ and $G_1 = \{2,3,4,5\}$. Two elements are chosen at random, without replacement, from the set $E_1$, and let $S_1$ denote the set of these chosen elements. Let $E_2 = E_1 \setminus S_1$ and $F_2 = F_1 \cup S_1$. Now two elements are chosen at random, without replacement, from the set $F_2$ and let $S_2$ denote the set of these chosen elements.
Let $G_2 = G_1 \cup S_2$. The value of $P(E_2 = F_2)$ is
(A) $\frac{1}{7}$
(B) $\frac{3}{7}$
(C) $\frac{1}{5}$
(D) $\frac{2}{7}$

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) $\frac { 1 } { 8 }$

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) $\frac { 5 } { 12 }$

A bag contains $N$ balls out of which 3 balls are white, 6 balls are green, and the remaining balls are blue. Assume that the balls are identical otherwise. Three balls are drawn randomly one after the other without replacement. For $i = 1,2,3$, let $W _ { i } , G _ { i }$, and $B _ { i }$ denote the events that the ball drawn in the $i ^ { \text {th } }$ draw is a white ball, green ball, and blue ball, respectively. If the probability $P \left( W _ { 1 } \cap G _ { 2 } \cap B _ { 3 } \right) = \frac { 2 } { 5 N }$ and the conditional probability $P \left( B _ { 3 } \mid W _ { 1 } \cap G _ { 2 } \right) = \frac { 2 } { 9 }$, then $N$ equals $\_\_\_\_$ .

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
$$P ( U ) = \frac { 1 } { 2 } , \quad P ( V ) = \frac { 1 } { 10 } , \quad \text { and } \quad P ( W ) = \frac { 1 } { 12 }$$
then $P ( T )$ is equal to

(A)

$\frac { 13 } { 36 }$

(B)

$\frac { 1 } { 3 }$

(C)

$\frac { 19 } { 60 }$

(D)

$\frac { 1 } { 4 }$

A factory has a total of three manufacturing units, $M _ { 1 } , M _ { 2 }$, and $M _ { 3 }$, which produce bulbs independent of each other. The units $M _ { 1 } , M _ { 2 }$, and $M _ { 3 }$ produce bulbs in the proportions of $2 : 2 : 1$, respectively. It is known that $20 \%$ of the bulbs produced in the factory are defective. It is also known that, of all the bulbs produced by $M _ { 1 } , 15 \%$ are defective. Suppose that, if a randomly chosen bulb produced in the factory is found to be defective, the probability that it was produced by $M _ { 2 }$ is $\frac { 2 } { 5 }$.
If a bulb is chosen randomly from the bulbs produced by $M _ { 3 }$, then the probability that it is defective is $\_\_\_\_$.

Let $A$ and $E$ be any two events with positive probabilities Statement I: $P ( E / A ) \geq P ( A / E ) P ( E )$. Statement II: $P ( A / E ) \geq P ( A \cap E )$.
(1) Both the statements are false
(2) Both the statements are true
(3) Statement-I is false, Statement-II is true
(4) Statement - I is true, Statement - II is false

A box $A$ contains 2 white, 3 red and 2 black balls. Another box $B$ contains 4 white, 2 red and 3 black balls. If two balls are drawn at random, without replacement from a randomly selected box and one ball turns out to be white while the other ball turns out to be red, then the probability that both balls are drawn from box $B$ is :
(1) $\frac { 7 } { 8 }$
(2) $\frac { 9 } { 16 }$
(3) $\frac { 7 } { 16 }$
(4) $\frac { 9 } { 32 }$

A box ' $A$ ' contains 2 white, 3 red and 2 black balls. Another box ' $B ^ { \prime }$ contains 4 white, 2 red and 3 black balls. If two balls are drawn at random, without replacement, from a randomly selected box and one ball turns out to be white while the other ball turns out to be red, then the probability that both balls are drawn from box ' $B ^ { \prime }$ is
(1) $\frac { 7 } { 16 }$
(2) $\frac { 9 } { 32 }$
(3) $\frac { 7 } { 8 }$
(4) $\frac { 9 } { 16 }$

Let $A$ and $B$ be two non-null events such that $A \subset B$. Then, which of the following statements is always correct?
(1) $P(A \mid B) \geq P(A)$
(2) $P(A \mid B) = P(B) - P(A)$
(3) $P(A \mid B) \leq P(A)$
(4) $P(A \mid B) = 1$

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 at least once is
(1) $\frac { 1 } { 4 }$
(2) $\frac { 1 } { 3 }$
(3) $\frac { 1 } { 8 }$
(4) $\frac { 1 } { 9 }$

In a group of 400 people, 160 are smokers and non-vegetarian; 100 are smokers and vegetarian and the remaining 140 are non-smokers and vegetarian. Their chances of getting a particular chest disorder are $35 \% , 20 \%$ and $10 \%$ respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is:
(1) $\frac { 14 } { 45 }$
(2) $\frac { 7 } { 45 }$
(3) $\frac { 8 } { 45 }$
(4) $\frac { 28 } { 45 }$

When a missile is fired from a ship, the probability that it is intercepted is $\frac { 1 } { 3 }$ and the probability that the missile hits the target, given that it is not intercepted, is $\frac { 3 } { 4 }$. If three missiles are fired independently from the ship, then the probability that all three hit the target, is:
(1) $\frac { 3 } { 8 }$
(2) $\frac { 1 } { 27 }$
(3) $\frac { 1 } { 8 }$
(4) $\frac { 3 } { 4 }$

A random variable $X$ has the following probability distribution:

$X$	0	1	2	3	4
$P ( X )$	$k$	$2 k$	$4 k$	$6 k$	$8 k$

The value of $P \left( \frac { 1 < x < 4 } { x \leq 2 } \right)$ is equal to
(1) $\frac { 4 } { 7 }$
(2) $\frac { 2 } { 3 }$
(3) $\frac { 3 } { 7 }$
(4) $\frac { 4 } { 5 }$

Let $A$ and $B$ be two events such that $P ( B \mid A ) = \frac { 2 } { 5 } , P ( A \mid B ) = \frac { 1 } { 7 }$ and $P ( A \cap B ) = \frac { 1 } { 9 }$. Consider $( S1 )\; P \left( A ^ { \prime } \cup B \right) = \frac { 5 } { 6 }$, $( S2 )\; P \left( A ^ { \prime } \cap B ^ { \prime } \right) = \frac { 1 } { 18 }$. Then
(1) Both $(S1)$ and $(S2)$ are true
(2) Both $(S1)$ and $(S2)$ are false
(3) Only $(S1)$ is true
(4) Only $(S2)$ is true

A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is
(1) $\frac{5}{3}$
(2) $\frac{2}{7}$
(3) $\frac{3}{7}$
(4) $\frac{5}{6}$

$25\%$ of the population are smokers. A smoker has 27 times more chances to develop lung cancer than a non-smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is $\frac { k } { 10 }$. Then the value of $k$ is $\_\_\_\_$.

In a bolt factory, machines $A , B$ and $C$ manufacture respectively $20 \% , 30 \%$ and $50 \%$ of the total bolts. Of their output 3, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found to be defective then the probability that it is manufactured by the machine $C$ is
(1) $\frac { 5 } { 14 }$
(2) $\frac { 9 } { 28 }$
(3) $\frac { 3 } { 7 }$
(4) $\frac { 2 } { 7 }$

Let Ajay will not appear in JEE exam with probability $p = \frac{2}{7}$, while both Ajay and Vijay will appear in the exam with probability $q = \frac{1}{5}$. Then the probability, that Ajay will appear in the exam and Vijay will not appear is:
(1) $\frac{9}{35}$
(2) $\frac{18}{35}$
(3) $\frac{24}{35}$
(4) $\frac{3}{35}$

A company has two plants $A$ and $B$ to manufacture motorcycles. $60\%$ motorcycles are manufactured at plant $A$ and the remaining are manufactured at plant $B$. $80\%$ of the motorcycles manufactured at plant $A$ are rated of the standard quality, while $90\%$ of the motorcycles manufactured at plant $B$ are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. Find the probability that it was manufactured at plant $B$.