Of the students in a school, it is known that $30\%$ have $100\%$ attendance and $70\%$ students are irregular. Previous years result report that $70\%$ of all students who have $100\%$ attendance attain A grade and $10\%$ irregular students attain A grade. At the end of the year, one student is chosen at random and he was found to have on A grade. What is the probability that the student has $100\%$ attendance?

It is known that $60\%$ students in a college reside in hostel while remaining $40\%$ are day scholars.$30\%$ of all students who reside in hostel attain A grade and $20\%$ of day scholars attain A grade. If one randomly chosen student has A grade, then what is the probability that he lives in the hostel?

A company has two plants to manufacture scooters. Plant I manufactures $80\%$  of the scooters and plant II manufactures $20\%$. At plant I, $85$ out of $100$  scooters are rated as of standard quality and at plant II, $65$ out of $100$ scooters are rated as of standard quality. A scooter is chosen at random and is found to be of standard quality. What is the probability that it has come from plant I?

In a class, having $60\%$ boys, $5\%$ of the boys and $10\%$ of the girls have an I.Q. of more than $150$. A student is selected at random and found to have an I.Q. of more than $150$. Find the probability that the selected student is a boy.

A man is known to speak truth $4$ out of $5$ times. He throws a die and reports that it is a six. Find the probability that it is actually a six.

Urn I has $2$ white and $3$ black balls, urn II has $4$ white and $1$ black ball and urn III has $3$ white and $4$ black balls. An urn is selected at random and a ball is drawn at random. What is the probability of drawing a white ball?