In Probability, we call `Paradox problems’ to those that can be apparently be solved by at least two different ways that lead to contradictory statements.

There is a large number of paradoxes in the history of Probability, which, properly introduced, can play a very useful role in teaching and learning activities, where formal methods abound. Therefore, several studies have investigated the use of paradoxes on students’ motivation of learning probability and statistics. They can serve as leverage to fruitful discussions, and provoke deeper thinking about the not always intuitive probabilistic concepts.

However, we must take into account that the use of paradoxes in the teaching can also carry potential danger, because they can cause a feeling of insecurity when the conflict between the mathematical solution and the intuition seems irresolvable. Thus, we cannot allow that they replace the theoretical learning.

In this work we analyze some well-known paradoxes (Monty Hall, the prisoners, Parrondo, St. Petersburg, birthday problem, the elevator, Lewis Carroll’s urn, …) examples of situations when our intuition guides us to a result but the mathematics tell us something quite different. They cover a range of techniques and concepts applicable to seminars or classes of varied levels. Our experience shows that students trend tend to use their previous intuitions rather than formal studies when they have to make a probabilistic judgment. When we introduce paradoxes in classroom, the students cannot remain passive and are forced to remove potential conflicts between intuition and theory.

In conclusion, although the main goal in teaching probability must not be just entertainment, we believe that paradoxes are a subject that has a lot of real life applications that should be discussed too.

[1] Batanero, C.; Henry, M. and Parzysz B. (2005). The nature of chance and probability. In Exploring probability in schools: Challenges for teaching and learning, ed. G. A. Jones , 15-37. New York: Springer.
[2] Lesser, L. (1998). Countering indifference using counterintuitive examples. Teaching Statistics, 20(1), 10-12.
[3] Leviatan, T. (1998). Misconceptions, fallacies, and pitfalls in the teaching of probability. ICM.
[4] López, L.; Montanero, J.; Mulero, M.A.; Parra, M.I. and Requejo B. (2015). Divulgationes Mathematicae. Badajoz: Cisan.
[5] Matthews, M. E. (2008). Selecting and using mathemagic tricks in the classroom. Mathematics Teacher, 102(2), 98-101.
[6] Shahvarani, Ahmad; Barahmand, Ali; Seif, Asghar (2012). The Effect of Employing Paradoxes in Teaching and Learning Mathematics in Correcting Students' Common Mistakes in Solving Equation. Life Science Journal-Acta Zhengzhou University Overseas Edition, 9(4), 5789-5792.
[7] Wilensky, U. (1995). Paradox, programming and learning probability: A case study in a connected mathematics framework. Journal of Mathematical Behavior, 14, 253-280.
keywords: paradox, probability.