A spacecraft is sent to observe Jupiter. It is supposed to take pictures of Jupiter's moons and send them back to Earth. There are three critical systems

involved: the camera, the batteries, and the transmission antenna. These three systems fail independently of each other. The mean lifetime of the battery system is 6 years, that of the camera is 12 years, and that of the antenna is 10 years. Assume all lifetimes are exponential random variables. The spacecraft is to reach Jupiter after 3 years to carry out this mission.

(a) What is the probability that the camera fails sooner than the transmission antenna?

(b) What is the probability that the mission is successful?

(c) Suppose the monitoring station on Earth conducts a test run after 2 years and finds that

no data are received, indicating that one of the three systems has failed. What is the

probability that the camera has failed?