Question

The electrical system for an assembly line has four components arranged in two parallel subsystems as shown in the diagram below. Each of the four components (labeled 1, 2, 3, 4) are independent and each has a 10% chance of failure. Because the components are independent, the two subsystems (labeled A and B) are also independent. If one component in a subsystem fails, then the whole subsystem fails because the power cannot get through. However, because there are two subsystems, only one subsystem needs to work to keep the assembly line running.

Let the random variable X = the number of subsystems that work.

(a) Make a table showing the probability distribution of X. (Show all calculations needed to compute the entries in the table.)

(b) Compute the expected number of subsystems that will work?

(c) What is the probability that the assembly line will keep running?

Answer 1

(a)

Here X can take value 0, 1, 2.

When X=0:

It means both subsytems fail. The subsytem has four components in series so it will not work is at least one component fail. The probabiltiy that subsystem A fail is

Likewise

So,

When X=1:  It means one subsytems fail. So,

When X=2:  It means both subsytems fail. So,

Following is the complete table:

X	P(X=x)
0	0.11826721
1	0.45126558
2	0.43046721

(b)

Following table shows the calculations:

X	P(X=x)	xP(X=x)
0	0.11826721	0
1	0.45126558	0.45126558
2	0.43046721	0.86093442
Total		1.3122

So, the expected number of subsystems that will work is

(c)

The probability that the assembly line will keep running is :