In a college,$25\%$ of the boys and $10\%$ of the girls offer Mathematics. The girls constitute $60\%$ of the total number of students. If a student is selected at random and is found to be studying Mathematics,the probability that the student is a girl,is

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 $III$?

$A$ bag contains four balls. Two balls are drawn randomly and found to be white. The probability that all the balls in the bag are white is

Two bags $A$ and $B$ contain $2$ white,$3$ black,$4$ red balls and $3$ white,$4$ black,$5$ red balls respectively. $A$ ball is drawn from bag $A$ and transferred to bag $B$. If a ball is then drawn from bag $B$,what is the probability that the ball drawn from bag $B$ is white?

An item is tested on a device for its defectiveness. The probability that such an item is defective is $0.3$. The device gives an accurate result in $8$ out of $10$ such tests. If the device reports that an item tested is not defective,then the probability that it is actually defective is:

$A$ diagnostic test has a probability of $0.95$ of giving a positive result when applied to a person suffering from a certain disease and a probability of $0.10$ of giving a positive result when given to a non-sufferer. It is estimated that $0.5 \%$ of the population are suffering from the disease. If this test is administered to a person from this population about whom there is no information relating to the incidence of this disease and the test gives a positive result, then the probability that the person is a sufferer is: