$A$ man speaks truth $2$ out of $3$ times. He picks one of the natural numbers in the set $S=\{1, 2, 3, 4, 5, 6, 7\}$ and reports that it is even. The probability that it is actually even 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?

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

Suppose that $5 \%$ of men and $0.25 \%$ of women have grey hair. $A$ person with grey hair is selected at random. What is the probability of this person being male? Assume that there are equal numbers of males and females.

$A$ bag contains $2n+1$ coins. It is known that $n$ of these coins have heads on both sides,whereas the other $n+1$ coins are fair. One coin is selected at random and tossed. If the probability that the toss results in heads is $\frac{31}{42}$,then the value of $n$ is

Bag $A$ contains $2$ white and $3$ red balls. Bag $B$ contains $4$ white and $5$ red balls. One bag is chosen at random and a ball is drawn from it,which is found to be red. Find the probability that it was drawn from bag $B$.