Two players flip a fair coin until either the sequence HTT...

Two players flip a fair coin until either the sequence HTT or HHT occurs. Player A wins if HTT occurs first, and player B wins if HHT occurs first. Show that player $\mathrm{B}$ is twice as likely to win.

Key Concepts

Markov Chains

Markov chains are used to model processes that move between a set of defined states, with the probability of each transition depending solely on the current state. In the context of sequential experiments such as coin tossing, the progression toward certain target patterns can be described by states that capture the history necessary to predict future outcomes.

Absorbing States

Absorbing states are particular states within a Markov chain that, once entered, cannot be left. They represent the termination of the process, such as achieving a designated pattern in a sequence, and are crucial for calculating the likelihood of various outcomes in sequence-based probability problems.

Recurrence Relations

Recurrence relations express probabilities in sequential processes by conditioning on what happens in the next step. They are used to construct equations that relate the probability of winning from a given state to the probabilities from subsequent states, enabling the systematic computation of hitting probabilities for different target sequences.

Overlapping Patterns

In problems involving sequential patterns, overlapping segments between potential target patterns can complicate analyses. Understanding how overlaps affect the transition probabilities between states is essential for correctly setting up the equations that lead to the final probabilities of each pattern occurring first.

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