A certain plant runs three shifts per day. Of all the items produced by the plant, 50% of them are produced on the first shift, 30% on the second shift, and 20% on the third shift. Of all the items produced on the first shift, 1% are defective, while 2% of the items produced on the second shift and 3% of the items produced on the third shift are defective.

a. An item is sampled at random from the day’s production, and it turns out to be defective. What is the probability that it was manufactured during the first shift?

b. An item is sampled at random from the day’s production, and it turns out not to be defective. What is the probability that it was manufactured during the third shift?