Question: A type of communications system works if at least half of its components work. Suppose it is possible to add a fifth component to such a system having four components. Show that the resulting five-component system is not necessarily more reliable. (Hint. In the notation of Example, it suffices to show that E4 ⊄ E5. Expressing E5 as the union of {at least three of the original four components work} and {two of the original four components work and the additional component works}, it can be seen that the event {two of the original four components work and the additional component does not work} is contained in E4 but not in E5.)
Example: The reliability of a system is defined as the probability that a system will function correctly under stated conditions. A system's reliability depends on the reliability of its components as well as the way the components are arranged. A type of communications system works if at least half of its components work. Suppose it is possible to add a sixth component to such a system having five components. Show that the resulting six-component system has improved reliability.