The Hubble space telescope carries six gyroscopes for its inertial navigation system. If fewer than three gyroscopes are operational, the telescope enters a sleep mode, and a shuttle mission is required to fix it. If all gyroscopes fail, the telescope will crash.
The gyroscope operating times, the time to reach sleep mode, and the time to prepare a shuttle mission are exponentially distributed with rates , , and , with units of inverse years. Integers are used here to represent the number of operational gyroscopes:
Find the probability that the telescope will crash in the next 10 years:
Find the probability that sleep mode is not reached (no shuttle mission is required) in 10 years: