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

$\mathbb P(A)=\mathbb P(A_1\cup\cdots\cup A_6)=\mathbb P(A_1)\times\cdots\times\mathbb P(A_6)$

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

$\mathbb P(A)=\mathbb P(A_1)^6=0.02^6=6.4\times10^{-11}=0.000000000064$