Question

A system with 6 independent components is such that the system will fail if at least one of the individual components fail. the probability that a component fails is 0.02. let a be the event that the system fails. what is the probability of a ?

Answer 1

The components operate independently, so the probability of "total" system failure is the same as the product of each probability of component failure. That is, if
`A_i` denotes the event that component
`i` fails, then


`\mathbb P(A)=\mathbb P(A_1\cup\cdots\cup A_6)=\mathbb P(A_1)*\cdots*\mathbb P(A_6)`

The
`A_i` are independent and identically distributed, so
`\mathbb P(A_i)=\mathbb P(A_j)` for all
`i,j`. So


`\mathbb P(A)=\mathbb P(A_1)^6=0.02^6=6.4*10^(-11)=0.000000000064`