$A$ bag contains $(N+1)$ coins: $N$ fair coins,and one coin with 'Head' on both sides. $A$ coin is selected at random and tossed. If the probability of getting 'Head' is $\frac{9}{16}$,then $N$ is equal to:

Urn $A$ contains $6$ red and $4$ black balls and urn $B$ contains $4$ red and $6$ black balls. One ball is drawn at random from urn $A$ and placed in urn $B$. Then one ball is drawn at random from urn $B$ and placed in urn $A$. If one ball is now drawn at random from urn $A$,the probability that it is found to be red,is

In a factory which manufactures bolts,machines $A, B$ and $C$ manufacture respectively $25\%, 35\%$ and $40\%$ of the bolts. Of their outputs,$5, 4$ and $2$ percent are respectively defective bolts. $A$ bolt is drawn at random from the product and is found to be defective. What is the probability that it is manufactured by the machine $B$?

The urns $A, B$ and $C$ contain $4$ red,$6$ black; $5$ red,$5$ black and $\lambda$ red,$4$ black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball drawn is red and the probability that it is drawn from urn $C$ is $0.4$,then find the square of the length of the side of the largest equilateral triangle inscribed in the parabola $y^2 = \lambda x$ with one vertex at the vertex of the parabola.

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 group of $400$ people,$160$ are smokers and non-vegetarian,$100$ are smokers and vegetarian,and the remaining $140$ are non-smokers and vegetarian. Their chances of getting a particular chest disorder are $35\, \%, 20 \,\%$,and $10 \,\%$ respectively. $A$ person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is ...... .