Ice machine 1 is currently working. Ice machine 2 will be put in use at a time $t$ from now. If the lifetime of ice machine $i$ is exponential with rate $\lambda_i$, $i = 1, 2$, by using hypo-exponential distribution:
a) What is the probability that ice machine 2 is the first machine to fail?
b) What is the probability that both ice machines will be fail at the same time?
I have problem to understand this question for (a) is this the same meaning that $$P(\text{Machine 2 fail first})= 1-P(\text{Machine 1 fail first})\\=1-\mathbb{P}(M_1 < M_2 + t) = 1-\mathbb{P}(M_1 - M_2 < t).$$ Also for (b) is this the same meaning of using the joint destiny function below? $$f(m_1,m_2) = f_{M_1}(m_1)\ f_{M_2}(m_2) = \lambda_1\lambda_2e^{-\lambda_1 m_1}e^{-\lambda_2 m_2},\qquad m_1,m_2 > 0.$$