# Probability of Failure in System

I was a given a question and its supposed solution, but I'm getting to a different result in the end. Please let me know if my reasoning is correct.

Consider the following system. The failure probability of each component is mentioned on the figure and we assume that the failures of the components are independent of each other. Assume that the failure probability of component A is equal to $$x$$. What is then the failure probability of the system (as a function of $$x$$)?

I reasoned it's easier to think of the probability of failure as $$1 - P\left( W \right)$$, where $$P(W)$$ is the probability that it works flawlessly. So, since in order for the system to work, each of the components must work, \eqalign{ & 1 - P\left( W \right) \cr & P\left( F \right) = 1 - P\left( W \right) = 1 - \left( {1 - 0.25} \right)\left( {1 - 0.1} \right)\left( {1 - 0.3} \right)\left( {1 - x} \right) \cr & P\left( F \right) = 1 - \left( {0.75} \right)\left( {0.9} \right)\left( {0.7} \right)\left( {1 - x} \right) \cr & P\left( F \right) = 0.5275 + 0.4725x \cr}

While the solution shows I experimented multiplying it to open the expression and it seems indeed a different one.

• The system will work if at least on one of the $3$ routes there is/are no failures. So not necessarily all routes. You better do not go for $1-P(W)$ here. – drhab Jan 28 '20 at 12:15

## 1 Answer

My assumption that all of them need to work was the problem; it turns out it's not a condition for the system to work.