Of the students in a college,it is known that $60 \%$ reside in a hostel and $40 \%$ are day scholars (not residing in a hostel). Previous year results report that $30 \%$ of all students who reside in a hostel attain $A$ grade and $20 \%$ of day scholars attain $A$ grade in their annual examination. At the end of the year,one student is chosen at random from the college and they have an $A$ grade. What is the probability that the student is a hostler?

$A$ factory has two machines $A$ and $B$. Past record shows that machine $A$ produced $60 \%$ of the items of output and machine $B$ produced $40 \%$ of the items. Further,$2 \%$ of the items produced by machine $A$ and $1 \%$ produced by machine $B$ were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by machine $B$?

$A, B, C$ are mutually exclusive and exhaustive events of a random experiment and $E$ is an event that occurs in conjunction with one of the events $A, B, C$. The conditional probabilities of $E$ given the happening of $A, B, C$ are respectively $0.6, 0.3$ and $0.1$. If $P(A)=0.30$ and $P(B)=0.50$,then $P(C \mid E)=$

In a certain recruitment test with multiple-choice questions, there are four options for each question, out of which only one is correct. An intelligent student knows $90 \%$ of the correct answers, while a weak student knows only $20 \%$ of the correct answers. If a weak student gets the correct answer, what is the probability that they were guessing?

Let $n_1$ and $n_2$ be the number of red and black balls,respectively,in box $I$. Let $n_3$ and $n_4$ be the number of red and black balls,respectively,in box $II$.
$1.$ One of the two boxes,box $I$ and box $II$,was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probability that this red ball was drawn from box $II$ is $\frac{1}{3}$,then the correct option$(s)$ with the possible values of $n_1, n_2, n_3$ and $n_4$ is(are):
$(A)$ $n_1=3, n_2=3, n_3=5, n_4=15$
$(B)$ $n_1=3, n_2=6, n_3=10, n_4=50$
$(C)$ $n_1=8, n_2=6, n_3=5, n_4=20$
$(D)$ $n_1=6, n_2=12, n_3=5, n_4=20$
$2.$ $A$ ball is drawn at random from box $I$ and transferred to box $II$. If the probability of drawing a red ball from box $I$,after this transfer,is $\frac{1}{3}$,then the correct option$(s)$ with the possible values of $n_1$ and $n_2$ is(are):
$(A)$ $n_1=4, n_2=6$
$(B)$ $n_1=2, n_2=3$
$(C)$ $n_1=10, n_2=20$
$(D)$ $n_1=3, n_2=6$
Give the answer for question $1$ and $2$.

Bag $A$ contains $3$ white and $5$ black balls, while bag $B$ contains $4$ white and $3$ black balls. $A$ ball is selected at random from bag $A$ and put into bag $B$. If a ball is now selected at random from bag $B$, what is the probability that this ball is white?