# Probability that system works for at least $t$ units of time

An electronic system contains two identical components which function independently of each other, but are connected in such a way that the system works if at least one of the components functions.

I am trying to find the probability that the two component system works for at least $t$ units of time assuming that the lifetime of each component follows an exponential distribution with mean $m$.

Let $X$ be the random variable denoting lifetime of each component of the system. Now I am confused whether to calculate $(P(X>t))^2$ or $(P(X>t))^2+2P(X>t)P(X<t)$ as the system works if at least one of the components work.

• It would be the second. Or perhaps more simply, $2P(X>t)-P(X>t)^2$ Apr 19 '18 at 17:01

Alternatively, the probability that the system $S$ works until at least time $t$ is: $P(S>t) = 1-P(S<t) = 1-P(X<t)*P(X<t) = 1-P(X<t)^2$