What is the probability that the system is operational? Consider a system of components in which there are 4 independent components, each of which possesses an operational probability of 0.9. The system does have a redundancy built in such that it does not fail if 3 out of the 4 components are operational. What is the probability that the total system is operational?  

I have no idea how to approach this. Can someone help?
The system fails when two, or more, components fail. If each component falls out independently (which is probably true) then we have a binomial distribution $Bin(0.9)$. Now it is easy to calculate the probability of system failure.  $$P( \text{failure}) = 1 - P(\text{operational}) = 1 - \left(P(\text{#fallout} = 0)+ P(\text{#fallout} = 1)\right)$$ 

$$ = 1-\left(\binom{4}{0}0.9^4 + \binom{4}{1}0.9^3\cdot0.1\right)$$
The system fails when two, or more, components fail. If each component falls out independently (which is probably true) then we have a binomial distribution $Bin(0.9)$. Now it is easy to calculate the probability of system failure.  $$P( \text{failure}) = 1 - P(\text{operational}) = 1 - \left(P(\text{#fallout} = 0)+ P(\text{#fallout} = 1)\right)$$ 

$$ = 1-\left(\binom{4}{0}0.9^4 + \binom{4}{1}0.9^3\cdot0.1\right)$$