Welcome to Math Help Forum!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.