A fair coin is flipped 10 times. Find the probability that...

A fair coin is flipped 10 times. Find the probability that there is a string of 4 consecutive heads by
(a) using the formula derived in the text;
(b) using the recursive equations derived in the text.
(c) Compare your answer with that given by the Poisson approximation.

Key Concepts

Recursive Methods in Sequence Analysis

Recursive techniques involve breaking down a complex counting or probability problem into simpler, related subproblems. In the context of sequences with pattern constraints, recursive equations can be formulated to determine the probability or count of sequences that meet the criteria (such as the occurrence of a specific consecutive pattern) by building the solution incrementally based on previous outcomes.

Bernoulli Trials and Pattern Occurrence

This concept involves analyzing sequences of independent experiments, such as coin flips, where each trial results in success or failure (heads or tails). A key aspect is determining the probability that a specific pattern (for example, a run of consecutive successes) occurs within the sequence. This requires understanding the structure of the sample space and identifying which outcomes satisfy the given conditions.

Poisson Approximation

The Poisson approximation is used to estimate the probability of a rare event occurring within a large number of trials. It simplifies calculations when the exact probability is small and the number of trials is high. By approximating the binomial distribution with a Poisson distribution under appropriate conditions, one can obtain estimates for the probability of events such as the occurrence of a specific run or pattern in a sequence.

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