In four schools $B_1, B_2, B_3, B_4$,the percentage of girl students is $12, 20, 13, 17$ respectively. From a school selected at random,one student is picked up at random and it is found that the student is a girl. The probability that the school selected is $B_2$ is:

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 $III$?

$A$ and $B$ are two events of a random experiment such that $P(B)=0.4$, $P(A \cap \bar{B})=0.5$, and $P(A \cup B) + P\left(\frac{B}{A \cup \bar{B}}\right) = 1.15$. Then $P(A) = $

Let $A, B,$ and $C$ be three mutually independent events. Consider the two statements $S_1$ and $S_2$:
$S_1 : A$ and $B \cup C$ are independent.
$S_2 : A$ and $B \cap C$ are independent.
Then:

An instructor has a question bank consisting of $300$ easy True / False questions,$200$ difficult True / False questions,$500$ easy multiple choice questions,and $400$ difficult multiple choice questions. If a question is selected at random from the question bank,what is the probability that it will be an easy question given that it is a multiple choice question?