Hello ntrantrinh

Welcome to Math Help Forum! Originally Posted by

**ntrantrinh** - a car manufacturer plant has three shifts working on the assembly line. The morning shift produces 38% of the total production; the afternoon shift produces 34%; and the evening shift 28%. Of their output 3%, 2%, 1%, respectively, do not pass quality control. If a vehicle is selected at random from the inventory and found defective, what is the probability that it was manufactured by:

a) morning shift (Given Answer: 0.543)

b) afternoon shift (Given Answer: 0.324)

c) evening shift (Given Answer: 0.133) - A small town has a network of 115 residential streets, all containing approximately the same number of residents. If a canvasser randomly selects 20 people from the phone book to promote a product, what is the probability that at least two of the people live on the same street?

(Given Answer: 0.827)

1) Bayes' theorem states that for two events $\displaystyle A$ and $\displaystyle B$:$\displaystyle p(A|B)=\frac{p(B|A)p(A)}{p(B)}$

where $\displaystyle p(A|B)$ is the conditional probability of $\displaystyle A$, given $\displaystyle B$; in other words, the probability that $\displaystyle A$ occurs, given that $\displaystyle B$ 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 say $\displaystyle A$ is the event 'the car is from the morning shift'; and

$\displaystyle B$ is the event 'the car is defective'

So we want $\displaystyle p(A|B)$.

Now $\displaystyle B|A$ is the event 'the car is defective, given that it is from the morning shift'. So $\displaystyle p(B|A) = 0.03$ (because 3% of the morning shift's output is defective).

And $\displaystyle p(A) = 0.38$ (I'm sure you can see why)

Now $\displaystyle p(B)$ the sum of the probabilities that a car chosen from a given shift is defective, which is $\displaystyle (0.38\times0.03)+(0.34\times0.02)+(0.28\times0.01) =0.021$.

So, using Bayes' theorem: $\displaystyle p(A|B)=\frac{0.03\times0.38}{0.021}=0.543$

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