An examination is attempted by $5000$ graduates, $2000$ post-graduates, and $1000$ doctorate holders. The probabilities that a graduate, a post-graduate, and a doctorate holder will pass the examination are $\frac{2}{3}$, $\frac{3}{4}$, and $\frac{4}{5}$ respectively. If one of the examinees passed the examination, then the probability that he is a post-graduate is:

$A$ factory has two machines $A$ and $B$. Past record shows that machine $A$ produced $60 \%$ of the items of output and machine $B$ produced $40 \%$ of the items. Further,$2 \%$ of the items produced by machine $A$ and $1 \%$ produced by machine $B$ were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by machine $B$?

Bag $A$ contains $6$ Green and $8$ Red balls and bag $B$ contains $9$ Green and $5$ Red balls. $A$ card is drawn at random from a well-shuffled pack of $52$ playing cards. If it is a spade,two balls are drawn at random from bag $A$,otherwise two balls are drawn at random from bag $B$. If the two balls drawn are found to be of the same colour,then the probability that they are drawn from bag $A$ is

$A$ student appears for a quiz consisting of only true-false type questions and answers all the questions. The student knows the answers of some questions and guesses the answers for the remaining questions. Whenever the student knows the answer of a question,he gives the correct answer. Assume that the probability of the student giving the correct answer for a question,given that he has guessed it,is $\frac{1}{2}$. Also assume that the probability of the answer for a question being guessed,given that the student's answer is correct,is $\frac{1}{6}$. Then the probability that the student knows the answer of a randomly chosen question is

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

An insurance company insured $2000$ scooter drivers,$4000$ car drivers,and $6000$ truck drivers. The probabilities of an accident are $0.01, 0.03,$ and $0.15$ respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?