Write a note on the concept of Martingales with illustrations.

The concept of Martingales is an important topic in probability theory and stochastic processes. It is a mathematical tool used to model and analyze random sequences of events. In simple terms, a Martingale is a sequence of random variables that exhibits a specific property called "martingale property." Let's explore this concept with some illustrations.

Consider a fair coin toss game, where you start with an initial amount of money. In each round, you bet a certain amount on the outcome of the coin toss. If the coin lands heads, you win the bet and receive double the amount you bet. If it lands tails, you lose the bet and your money is reduced to half. Let's _____ ___ _____ _____ ___ _________ ___ ___ ____.
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