Hello ntrantrinh

Welcome to Math Help Forum!1) Bayes' theorem states that for two events and :where is the conditional probability of , given ; in other words, the probability that occurs, given that has occurred; etc...

In part (i), we want the probability that a car chosen at random is from the morning shift, given that it is defective. So let's sayis the event 'the car is from the morning shift'; andSo we want .

is the event 'the car is defective'

Now is the event 'the car is defective, given that it is from the morning shift'. So (because 3% of the morning shift's output is defective).

And (I'm sure you can see why)

Now the sum of the probabilities that a car chosen from a given shift is defective, which is .

So, using Bayes' theorem:Parts (ii) and (iii) follow in the same way.

I don't have time right now to look at your second question, but I'll do so later unless someone else answers it first.

Grandad