A circuit has three major components, $A, B$ and $C$. A component $A$ operates independently of $B$ and $C$, the components $B$ and $C$ are interdependent. It is known that the component $A$ works properly $70\%$ of the time, component $B$$ 85\%$ of the time and $C$$99\%$ of the time. However, if component C fails, there is a $55\%$ chance that B will also fail. Assume that at least two parts must operate for the circuit to function. What is the probability that the circuit will work properly?
I think it is $0.7\times0.85\times0.99 + 0.7\times0.45\times0.01 + 0.7\times0.99\times0.15 + 0.85\times0.99\times0.3$ (all of them working $+ A$ and $B$ working $+ A$ and $C$ working $+ B$ and $C$ working )
Is it okay ? thank you