Parrondo's paradox was introduced by Juan Parrondo in 1996. In game theory, this paradox is described as: A combination of losing strategies becomes a winning strategy. At first glance, this paradox is quite surprising, but we can easily explain it by using simulations and mathematical arguments. Indeed, we first consider some examples with the Parrondo's paradox and, using the software R, we simulate one of them, the coin tossing. Actually, we see that specific combinations of losing games become a winning game. Moreover, even a random combination of these two losing games leads to a winning game. Later, we introduce the major definitions and theorems over Markov chains to study our Parrondo's paradox applied to the coin tossing problem. In particular, we represent our Parrondo's game as a Markov chain and we find its stationary distribution. In that way, we exhibit that our combination of two losing games is truly a winning combination. We also deliberate possible applications of the paradox in some fields such as ecology, biology, finance or reliability theory.

翻译：帕龙多悖论由Juan Parrondo于1996年提出。在博弈论中，该悖论描述为：若干失败策略的组合反而成为获胜策略。乍看之下，这一悖论令人惊讶，但通过模拟和数学论证可轻易解释。我们首先考虑帕龙多悖论的若干实例，并利用R软件对其中的抛硬币问题进行模拟。实际上，我们发现特定组合的失败游戏会演变为获胜游戏。更令人惊奇的是，即使随机组合这两个失败游戏也能产生获胜结果。随后，我们引入马尔可夫链的主要定义与定理，以研究应用于抛硬币问题的帕龙多悖论。特别地，我们将帕龙多游戏建模为马尔可夫链并求其平稳分布，从而证明两个失败游戏的组合确实构成获胜组合。最后，我们探讨了该悖论在生态学、生物学、金融学或可靠性理论等领域的可能应用。