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?

There are $2$ bags each containing $3$ white and $5$ black balls and $4$ bags each containing $6$ white and $4$ black balls. If a ball drawn randomly from a bag is found to be black,then the probability that this ball is from the first set of bags is

Bag $A$ contains $2$ white and $3$ red balls and bag $B$ contains $4$ white and $5$ red balls. If one ball is drawn at random from one of the bags and is found to be red,then the probability that it was drawn from the bag $B$ is

$A$ 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

In a certain recruitment test with multiple choice questions,there are four options to answer 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 an intelligent student gets the correct answer for a question,what is the probability that he was guessing it?