In this dissertation we present the definition of the expected value of a random variable, an important probabilistic concept which is useful in many applications but which, for historical reasons, is not taught in high school in Brazil. We also discuss examples of interesting and challenging probability problems, including questions from some of the hardest exams in the country, such as the Vestibular for the Instituto Militar de Engenharia (IME) and the Desafio em Matemática of PUC-Rio. In many of the proposed activities, we use computational tools such as Excel and GeoGebra: these can become allies when teaching probability and help in the understanding of the concept of expected value.