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) $\frac { 2 } { 3 }$