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, $$\eqalign{ & 1 - P\left( W \right) \cr & P\left( F \right) = 1 - P\left( W \right) = 1 - \left( {1 - 0.25} \right)\left( {1 - 0.1} \right)\left( {1 - 0.3} \right)\left( {1 - x} \right) \cr & P\left( F \right) = 1 - \left( {0.75} \right)\left( {0.9} \right)\left( {0.7} \right)\left( {1 - x} \right) \cr & P\left( F \right) = 0.5275 + 0.4725x \cr} $$
I experimented multiplying it to open the expression and it seems indeed a different one.