Final answer:
The complement of Mariah choosing three mysteries is her not choosing all mysteries, which means selecting any combination that includes at least one biography or science fiction novel. The sample space consists of all possible three-book combinations excluding all mystery combos. The sum of the probabilities of an event and its complement is always 1.
Step-by-step explanation:
Mariah is randomly choosing three books to read from a selection that includes 5 mysteries, 7 biographies, and 8 science fiction novels. The complement of Mariah choosing three mysteries is not choosing all mysteries. When the problem refers to the complement of an event, it is talking about all possible outcomes that do not include the event itself. Hence, Mariah can choose any combination of books as long as they are not all mysteries.
The sample space for Mariah choosing three books from the shelf is the set of all possible combinations of three books from the total of 20 books. However, when we are talking about the complement of choosing three mysteries, the sample space gets reduced to combinations that include at least one biography or science fiction novel.
The sum of the probabilities of an event and its complement is always 1. This is because between an event and its complement, all possible outcomes are covered.