1. The problem statement, all variables and given/known data A submarine has three navigational devices but can remain at sea if at least two are working. Suppose that the failure times are exponential with means 1 year, 1.5 years, and 3 years. What is the average length of time the boat can remain at sea? 2. Relevant equations Density for an exponential random variable: f(x)= λe-λx for x>=0 E(T)=1/λ if T is an exponential random variable Maybe relevant: P(S<T)=λS / (λS + λT) for exponential random variables S and T, and this is similar for many exponential random variables. 3. The attempt at a solution The boat can remain at sea until 2 parts break. Let T be the time that the boat can remain at sea. Since the sample space can be divided into the order in which the parts fail, we have: E(T)= Sum where 1<=i,j,k<=3 of: E(T|Ti<Tj<Tk)P(Ti<Tj<Tk) where E(T|Ti<Tj<Tk)=E(Tj|Ti<Tj<Tk) since the boat can remain at sea until two parts fail. Now, this would be a similar method that we've used with discrete random variables in class. Unfortunately, I'm a little rusty with my continuous probability. Is there somewhere to go from here, or maybe an easier way to approach the problem? I very much appreciate any help you can give! Thanks!