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

$A$ box contains $4$ black,$2$ white,and $6$ red balls. Another box contains $3$ black and $5$ white balls. An unbiased die is thrown. If $1$ or $2$ appears on the die,a ball is drawn from the first box; otherwise,a ball is drawn from the second box. If the drawn ball is black,what is the probability that $2$ appeared on the die?

The probabilities of having a defective toy in three cartons $A, B, C$ are $\frac{1}{3}, \frac{1}{4}, \frac{2}{5}$ respectively. If a carton is selected at random and a toy drawn randomly from it is found to be defective,then the probability that it is drawn from carton $B$ 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:

Bag $A$ contains $3$ white and $7$ red balls,and bag $B$ contains $3$ white and $2$ red balls. One bag is selected at random and a ball is drawn from it. What is the probability that the ball was drawn from bag $A$,given that the ball drawn 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: