6. Suppose two components are connected in a series, so that the whole system wo

Question

6. Suppose two components are connected in a series, so that the whole system works if and only if both components work. Each component consists of two units connected in parallel, so that the component works as long as either unit works. Thus we have 2 components attached to each other (both must work), with each component comprising of 2 units (at least 1 unit must work for single component to work). Assume all units are independent and fail with probability p (work with probability 1 p). Also assume the components are independent of each other. What is the probability the whole system works? Hint: The probability a single component will work is equal to one minus the probability of the component not working (which occurs when both units within component dont work).

Explanation / Answer

P(Whole system works)

= P(First component work)*P(Second component work)

= P(Atleast one unit of first component work)*P(Atleast one unit of second component work)

= (1 - P(No unit of first component work))*(1 - P(No unit of second component work))

= (1 - (1 - p)2)*(1 - (1 - p)2)

= (2p - p2)2

= 4p2 + p4 - 4p3

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