5. An electrical system consists of five components as...

5. An electrical system consists of five components as illustrated in Figure 1. The system works if either the components A and B work or the components C, D, and E work. The reliability (probability of working) of each component is also shown in Figure 1. Assume the components fail independently. Figure 1: Diagram for Problem 5. [5] (a) What is the probability that the entire system works? [5] (b) Given that the system works, what is the probability that component A is not working? 6. In a certain assembly plant, three machines, B1, B2, and B3, make 30%, 45%, and 25%, respectively, of the products. It is known from past experience that 2%, 3%, and 2% of the products made by each machine, respectively, are defective. [5] (a) If a finished product is randomly selected, what is the probability that it is defective? [5] (b) If a product was chosen randomly and found to be defective, what is the probability that it is made by machine B3?

Transcript

00:01 So the first part of this problem, we're given the following machine circuit type thing that is functional if either a and b work or if c, d, and e work.

00:11 So it needs only one of these paths for the entire thing to be functional.

00:16 So the first question says, what is the probability that the machine is functional? and these are the individual probabilities that each component is functional.

00:23 So the probability that the machine is functional is equal to the probability that the that a and b are functional, or that c and d and e are functional.

00:50 So the probability, and just generally speaking, the probability that a or b, and these are general events, a or b, is equal to the probability of a plus the probability of b minus the probability of a and b.

01:12 So first we need to find the probability of a and b working on their own independently.

01:23 So this is just going to be, because they're independent, we can say that the probability of a and b is equal to the probability of a times the probability of b because they're independent.

01:37 So in this case, it's equal to 0 .7 times 0 .49.

01:43 Similarly, i'm going to go to a new page.

01:47 Probability of c and d and e working is equal to 0 .8 times 0 .8 times 0 .8 cubed, which is equal to 0 .512.

02:14 Now i need to figure out what the probability of both of these events occurring is to subtract.

02:20 And so for both of these events to occur, it's going to be, okay, so let's write this as a and b and c and d and e is just equal to every single one of them working, which is equal to the probabilities, again, because these are all independent of them all multiplied together, which is equal to 0 .7 squared times 0 .8 cubed, is equal to 0 .49 times 0 .512, which is equal to 0 .2588.

03:16 So the probability that the machine functions is equal to 0 .49 plus 0 .512 minus 0 .208.

03:30 And this comes out to equal .75112.

03:40 So now we're asked, given that the machine is functional, what is the probability that a is not working given that the machine is functional? and so the probability of a given b is equal to the probability of a and b divided by the probability of b.

04:12 And so we know that if a is not working, that for the machine to be functional, c, d, and e have to be working.

04:25 So the probability of a not working and the machine being functional is equal to the probability of a not working times the probability of c, probability of d, and the probability of d, and the probability of the probability of the probability.

04:46 Probability of e all working.

04:49 So the probability that a doesn't work is 1 minus 0 .7 or 0 .3.

04:58 And all of these, like we said, is equal to 0 .8 cubed.

05:02 And so we have 0 .8 times 0 .8 times 0 .3...

Question

Please give Ace some feedback