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 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 $\_\_\_\_$.

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) $\frac { 9 } { 16 }$

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) $\frac{5}{6}$

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) $\frac { 2 } { 7 }$

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 $B$.

Bag $A$ contains 3 white, 7 red balls and bag $B$ contains 3 white, 2 red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from the bag $A$, if the ball drawn is white, is:
(1) $\frac{1}{4}$
(2) $\frac{1}{9}$
(3) $\frac{1}{3}$
(4) $\frac{3}{10}$

There are three bags $X , Y$ and $Z$. Bag $X$ contains 5 one-rupee coins and 4 five-rupee coins; Bag $Y$ contains 4 one-rupee coins and 5 five-rupee coins and Bag Z contains 3 one-rupee coins and 6 five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability, that it came from bag Y , is : (1) $\frac { 1 } { 4 }$ (2) $\frac { 1 } { 2 }$ (3) $\frac { 5 } { 12 }$ (4) $\frac { 1 } { 3 }$

Bag $B _ { 1 }$ contains 6 white and 4 blue balls, Bag $B _ { 2 }$ contains 4 white and 6 blue balls, and Bag $B _ { 3 }$ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag $B _ { 2 }$, is :
(1) $\frac { 4 } { 15 }$
(2) $\frac { 1 } { 3 }$
(3) $\frac { 2 } { 5 }$
(4) $\frac { 2 } { 3 }$

Q80. Three urns A, B and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn A is :
(1) $\frac { 5 } { 18 }$
(2) $\frac { 5 } { 16 }$
(3) $\frac { 4 } { 17 }$
(4) $\frac { 7 } { 18 }$

Q80. 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. If $p$ is the probability that it was manufactured at plant $B$, then $126 p$ is
(1) 54
(2) 66
(3) 64
(4) 56
Q81.Let $x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 }$ be the solution of the equation $4 x ^ { 4 } + 8 x ^ { 3 } - 17 x ^ { 2 } - 12 x + 9 = 0$ and $\left( 4 + x _ { 1 } ^ { 2 } \right) \left( 4 + x _ { 2 } ^ { 2 } \right) \left( 4 + x _ { 3 } ^ { 2 } \right) \left( 4 + x _ { 4 } ^ { 2 } \right) = \frac { 125 } { 16 } m$. Then the value of $m$ is

Q80. There are three bags $X , Y$ and $Z$. Bag $X$ contains 5 one-rupee coins and 4 five-rupee coins; Bag $Y$ contains 4 one-rupee coins and 5 five-rupee coins and Bag Z contains 3 one-rupee coins and 6 five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability, that it came from bag Y , is :
(1) $\frac { 1 } { 4 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 5 } { 12 }$
(4) $\frac { 1 } { 3 }$