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:

In an entrance test, there are multiple-choice questions. Each question has four possible answers, only one of which is correct. The probability that a student knows the answer to a question is $90\%$. If the student gets the correct answer to a question, what is the probability that they were guessing?

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:

$A$ factory has a total of three manufacturing units,$M_1, M_2$,and $M_3$,which produce bulbs independently. The units $M_1, M_2$,and $M_3$ produce bulbs in the proportions of $2: 2: 1$,respectively. It is known that $20\%$ of the bulbs produced in the factory are defective. It is also known that,of all the bulbs produced by $M_1, 15\%$ are defective. Suppose that,if a randomly chosen bulb produced in the factory is found to be defective,the probability that it was produced by $M_2$ is $\frac{2}{5}$. If a bulb is chosen randomly from the bulbs produced by $M_3$,then the probability that it is defective is $.....$ .

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:

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?