Question

Assume c and d function independently. for the system to function, either c or d must function.

Answer 1

For the "system" to function, we require that $C\cup D$ occurs (i.e. the event that either $C$ or $D$ function properly). This occurs with probability

$\mathbb P(C\cup D)=\mathbb P(C)+\mathbb P(D)-\mathbb P(C\cap D)$

but since $C$ and $D$ are independent,

$\mathbb P(C\cup D)=\mathbb P(C)+\mathbb P(D)-\mathbb P(C)\mathbb P(D)$

We're given that $\mathbb P(C^C)=0.08$ and $\mathbb P(D^C)=0.12$, so

$\mathbb P(C\cup D)=(1-0.08)+(1-0.12)-(1-0.08)(1-0.12)=0.9904$

If $\mathbb P(C^C)=\mathbb P(D^C)=p$, then

$\mathbb P(C\cup D)=(1-p)+(1-p)-(1-p)^2=1-p^2$