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?

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

A fair die is rolled. If face $1$ turns up, a ball is drawn from bag A. If face $2$ or $3$ turn up, a ball is drawn from bag B. If face $4$ or $5$ or $6$ turn up, a ball is drawn from bag C. Bag A contains $3$ red and $2$ white balls, bag B contains $3$ red and $4$ white balls and bag C contains $4$ red and $5$ white balls. The die is rolled, a bag is picked up and a ball is drawn. If the ball drawn is red, what is the probability that it is drawn from bag B?

Each of three identical jewellery boxes have two drawers. In each drawers of the first box there is a gold watch. In each drawers of the second box there is a silver watch. In one drawer of the third box there is a gold watch while in the other there is a silver watch. If we select a box at random, open one of the drawers and find it containing a silver watch, then what is the probability that the other drawer has a gold watch?

A factory has three machines A, B and C producing $1500, 2500 \mathrm{and} 3000$ bulbs per day respectively. Machine A produces $1.5\%$ defective bulbs, machine B produces $2\%$  defective bulbs and machine C produces  $2.5\%$ defective bulbs. At the end of the day, a bulb is drawn at random and it is found to be defective. What is the probability that this defective bulb has been produced by machine B?