A and B flip coins. A starts and continues flipping until a...

$A$ and $B$ flip coins. $A$ starts and continues flipping until a tail occurs, at which point $B$ starts flipping and continues until there is a tail. Then $A$ takes over, and so on. Let $P_{1}$ be the probability of the coin's landing on heads when $A$ flips and $P_{2}$ when $B$ flips. The winner of the game is the first one to get (a) 2 heads in a row; (b) a total of 2 heads; (c) 3 heads in a row; (d) a total of 3 heads. In each case, find the probability that $A$ wins.

$A$ and $B$ flip coins. $A$ starts and continues flipping until a tail occurs, at which point $B$ starts flipping and continues until there is a tail. Then $A$ takes over, and so on. Let $P_{1}$ be the probability of the coin's landing on heads when $A$ flips and $P_{2}$ when $B$ flips. The winner of the game is the first one to get
(a) 2 heads in a row;
(b) a total of 2 heads;
(c) 3 heads in a row;
(d) a total of 3 heads. In each case, find the probability that $A$ wins.

Key Concepts

Turn-Based Alternating Dynamics

This describes the framework where two players alternate turns based on specific triggers (e.g., flipping a tail leads to switching players). The analysis must account for the fact that each player may have different probabilities when flipping the coin, and the switching mechanism affects the transition dynamics between states, which is essential for computing the win probabilities.

Sequential versus Cumulative Win Conditions

This concept differentiates between win conditions that require consecutive outcomes (such as achieving a specific number of heads in a row) and those that count the total occurrence of an event (like a cumulative number of heads). Each type of condition has unique probabilistic considerations and influences the structure of the state space and the recurrence formulas used in the analysis.

Markov Chain Modeling

This concept involves representing the game as a series of states with transitions determined by the outcome of coin flips. In such a model, the future state depends only on the present state (the memoryless property), which is key for evaluating probabilities in sequential games. Using a Markov chain framework allows for systematically analyzing the progression of the game toward winning conditions.

Absorbing States

In the context of sequential games, absorbing states are the terminal states where the game ends, such as when a player achieves the winning condition. Understanding absorbing states is crucial because they represent outcomes that stop further transitions in the chain, and calculating the absorption probabilities helps determine the overall chance of winning for each player.

Recurrence Relations and Conditional Probabilities

These tools are used to set up equations that relate the probability of winning from one state to probabilities from other states, based on the outcome of each coin flip. Recurrence relations account for the fact that future events depend on the current outcome, and conditional probabilities allow for the division of complex events into simpler, analyzable components.

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