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The concept of Expected Value plays a crucial role when dealing with probabilities, like the chance of a coin landing heads. In simple terms, Expected Value (denoted as \( E[X] \)) is the average or mean value you would anticipate over a large number of trials. It's a way to predict what to expect in the long run.
For example, when you flip a fair coin, which has equal probability of heads or tails, the expected number of heads you get is around half of the number of flips.
In the context of this exercise, the expected value \( E[N] \) represents the average number of coin flips required until two out of the last three are heads. This gives us insight into how many attempts are usually needed, on average, emphasizing the significance of expected value in predicting outcomes in random processes.

Markov Chains serve as a mathematical system that undergoes transitions between states, following certain probabilistic rules. Markov Chains are "memoryless," which means that the next state depends solely on the current state and not on the sequence of events prior to it. This is a property known as the "Markov property."
In the coin flipping exercise, the states can be the sequences of heads and tails in the recent flips. Each flip transitions from one state to another, and these transitions can be described with probabilities. For example, from a state of 'one head in the last two flips', there are probabilities to flip another head or tails, transitioning to states with 'two heads' or 'returns to zero heads'.
This concept provides a structured way to model and analyze random processes where the future is independent of the past, once you know the present. By leveraging Markov Chains, we can understand the transitions of getting a certain sequence of heads and, ultimately, the expectation of steps to reach the condition required.

Stochastic Processes encompass collections of random variables that evolve over time, often used to model uncertainties and systems evolving randomly. Each state or event is part of a set of possible outcomes, and stochastic processes help in predicting future probabilities based on randomness and inherent unpredictability.
In terms of coin flipping, this exercise is a type of stochastic process where each flip of the coin is a random event influenced by probability \( p \), the likelihood of landing on heads. As you continue to flip the coin, you observe a sequence of random outcomes that form the stochastic process, which could be visualized through a flow of random events over time.
This concept is valuable in analyzing how the coin transitions among different states of heads and tails configurations over various trials. It provides a broad framework under which Markov Chains are specialized to capture these random transitions. Understanding stochastic processes can significantly enhance insights into systems and phenomena dictated by random occurrences.