In designing a new space vehicle, NASA needs to decide whether to provide 0, 1 or 2 backup systems for a critical component of the vehicle. The first backup system, if included, comes into use only if the original system fails. The second backup system, if included, comes into use only if the original system and the first backup system both fail. NASA engineers claim that each system, independent of the others, has a 5% chance of failing if called into use. Each backup system costs $35,000 to produce and install within the vehicle. Once the vehicle is in flight, the mission will be scrubbed (ended early) only if the original system and all backups fail. The cost of a scrubbed mission, in addition to production costs, is assessed to be $10,000,000.
How many backup systems are needed to minimize NASA's expected total cost?