A system composed of n separate components is said to be a p

A system composed of n separate components is said to be a parallel system if it functions when at least one of the components functions. Suppose the following systems function if current flows from A to B. If each switch (break in the line) is activated (i.e. closed to allow electricity to flow across the line) independently with probability p = 0.3, what is the probability the system functions? Solve for each circuit.

c) For the first circuit, (a), find the probability that at least 3 of the switches are activated.

Solution

1. For the first circuit: Probability that the system functions is : 1 - probability that none functions = 1- 0.7^4 = 0.7599

   For the second circuit: Probabilty that current flows at the 1st branch = 0.3 x 0.3 = 0.09

                                     Probabilty that current flows at the 2nd branch = 0.3

                                     Probabilty that current flows at the 3rd branch = 0.3 x 0.3 x 0.3 = 0.027

                                     Probabilty that current flows through the system = 1 - {(1-0.09)(1-0.3)(1-0.027)} = 0.3802

2. Probabilty that atleast three switches are activated : Probaility that three switches are working & one not working + Probability that all switches are working = 0.3 x 0.3 x 0.3 x 0.7 + 0.3 ^ 4 = 0.1971