Question

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.

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?

Answer 1

The probability that the camera has failed given that one of the systems has failed after 2 years can be calculated using the conditional probability formula and the reliability functions for exponential random variables.

Step-by-step explanation:

To calculate the probability that the camera has failed given that one of the three systems (camera, batteries, antenna) on the spacecraft has failed after 2 years, we should use the conditional probability formula and the properties of exponential random variables.

The reliability function for an exponential random variable with a mean lifetime μ is R(t) = e-t/μ. Thus:

• Reliability of the battery after 2 years: RB(2) = e-2/6
• Reliability of the camera after 2 years: RC(2) = e-2/12
• Reliability of the antenna after 2 years: RA(2) = e-2/10

The probability of failure for each system is 1 minus its reliability. Given that one of the systems has failed, the conditional probability that it was the camera is:

P(Camera failed | One system failed) = P(Camera failed) / (P(Camera failed) + P(Battery failed) + P(Antenna failed))

Using the reliability functions to find the failure probabilities and plugging them into the formula, we can calculate the desired probability.

The result provides the likelihood that the camera was the component that failed, given that a failure was detected during the test run.