Question

A parallel system functions whenever at least one of its components works. Consider a parallel systemofncomponents, and suppose that each component works independently with probability 1/2. Find theconditional probability that component 1 works given that the system is functioning.

Answer 1

Answer:

The conditional probability that component 1 works given that the system is functioning =  $\frac{\frac{1}{2} }{1 - (\frac{1}{2})^{n} }$ .

Step-by-step explanation:

We are given that a parallel system functions whenever at least one of its components works.

There are parallel system of n components and each component works independently with probability 1/2.

Let A = Probability of component 1 working properly, P(A1) = 1/2 = 0.5

Also let S = Probability that system is functioning for whole n components, P(S)

Now, the conditional probability that component 1 works given that the system is functioning is given by P(A1/S) ;

 P(A1/S) = $\frac{P(A1\bigcap S)}{P(S)}$ {Means P(component 1 working and system also working)

                                            divided by P(system is functioning)

  P(A1/S) = $\frac{P(A1)}{1-P(S)}$ {In numerator it is P(component 1 working) and in

                                denominator it is P(system not working) = 1 - P(system is

                                                                                                             working) }

Since we know that P(system not working) means that none of the components is working in system and it is given with the probability of 0.5 and since there are total of n components so P(system not working) = 1 - $(\frac{1}{2})^{n}$ .

 Hence, P(A1/S) = $\frac{\frac{1}{2} }{1 - (\frac{1}{2})^{n} }$ .