A person continually flips a coin until a run of 3...

A person continually flips a coin until a run of 3 consecutive heads appears. Assuming that each flip independently lands heads with probability $p$, determine the expected number of flips required. HINT: Let $T$ denote the first flip that lands on tails and let it be 0 if all flips land on heads, and then condition on $T$.

A person continually flips a coin until a run of 3 consecutive heads appears. Assuming that each flip independently lands heads with probability $p$, determine the expected number of flips required.
HINT: Let $T$ denote the first flip that lands on tails and let it be 0 if all flips land on heads, and then condition on $T$.

Key Concepts

Stopping Time

Stopping time refers to a random variable that signifies the time at which a certain event occurs for the first time. In the analysis of coin tosses, it is used to denote the number of flips taken to achieve a pre-defined condition, such as obtaining a run of consecutive heads. This concept is key in evaluating the process and calculating expected durations.

Coin Toss Experiment

This concept involves performing independent Bernoulli trials where each flip of a coin has a preset probability of success (heads) or failure (tails). It forms the basis of modeling random events with a fixed probability, which is crucial in determining outcomes and analyzing sequential events in probability theory.

Expected Value

Expected value is a measure of the average outcome of a random experiment if it were repeated many times. In the context of sequential events, such as repeatedly flipping a coin until a specified sequence occurs, it represents the average number of flips required to achieve the desired outcome.

Conditional Expectation

Conditional expectation is a fundamental tool in probability that involves computing the expected value of a random variable given that a certain event has occurred. It helps break down complex problems into simpler subproblems by conditioning on an event, thus simplifying the analysis of sequential random processes.

Markov Chain and Recurrence Relations

A Markov chain is a stochastic model describing a sequence of possible events where the probability of each event depends only on the state attained in the previous event. Recurrence relations can be derived from this structure, providing equations that recursively relate the expected value in terms of previous states. This is particularly useful when analyzing sequences or runs within the context of repeated coin tosses.

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