Given three identical boxes $I$,$II$ and $III$,each containing two coins. In box $I$,both coins are gold coins,in box $II$,both are silver coins and in the box $III$,there is one gold and one silver coin. $A$ person chooses a box at random and takes out a coin. If the coin is of gold,what is the probability that the other coin in the box is also of gold?

$A$ bag $A$ contains $2$ white and $3$ red balls and bag $B$ contains $4$ white and $5$ red balls. One ball is drawn at random from a randomly chosen bag and is found to be red. The probability that it was drawn from bag $B$ is:

If a machine is correctly set up,it produces $90\%$ acceptable items. If it is incorrectly set up,it produces only $40\%$ acceptable items. Past experience shows that $80\%$ of the set ups are correctly done. If after a certain set up,the machine produces $2$ acceptable items,find the probability that the machine is correctly set up.

Coloured balls are distributed in four boxes as shown in the following table:

Box	Black	White	Red	Blue
$I$	$3$	$4$	$5$	$6$
$II$	$2$	$2$	$2$	$2$
$III$	$1$	$2$	$3$	$1$
$IV$	$4$	$3$	$1$	$5$

$A$ box is selected at random and then a ball is randomly drawn from the selected box. If the colour of the ball is black,what is the probability that the ball drawn is from box $III$?

Suppose that $5 \%$ of men and $0.25 \%$ of women have gray hair. $A$ person with gray hair is selected at random. If there are an equal number of males and females,then the probability that the selected person is a man is:

The urns $A, B$ and $C$ contain $4$ red,$6$ black; $5$ red,$5$ black and $\lambda$ red,$4$ black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball drawn is red and the probability that it is drawn from urn $C$ is $0.4$,then find the square of the length of the side of the largest equilateral triangle inscribed in the parabola $y^2 = \lambda x$ with one vertex at the vertex of the parabola.