Independent flips of a coin that lands on heads with...

Independent flips of a coin that lands on heads with probability $p$ are made. What is the probability that the first four outcomes are (a) $H, H, H, H ?$ (b) $T, H, H, H ?$ (c) What is the probability that the pattern $T, H, H, H$ occurs before the pattern $H, H, H, H ?$ Hint for part $(c):$ How can the pattern $H, H, H, H$ occur first?

Independent flips of a coin that lands on heads with probability $p$ are made. What is the probability that the first four outcomes are
(a) $H, H, H, H ?$
(b) $T, H, H, H ?$
(c) What is the probability that the pattern $T, H, H, H$ occurs before the pattern $H, H, H, H ?$

Hint for part $(c):$ How can the pattern $H, H, H, H$ occur first?

Key Concepts

Independent Events

This concept refers to the idea that the outcome of one coin flip does not affect the outcome of another. In probability theory, when events are independent, the joint probability of multiple events occurring is the product of their individual probabilities.

Multiplication Rule in Probability

The multiplication rule is used to calculate the probability of a sequence of independent events occurring in a specific order by multiplying the probability of each event. For instance, the probability of a specific sequence of coin flips is obtained by multiplying the probability of each flip outcome.

Sequential Pattern Matching

This concept involves determining the likelihood of a particular ordered sequence of outcomes occurring in an independent sequence of trials. It is critical when identifying the probability of observing specific patterns, such as strings of heads and tails in coin tosses.

Competing Patterns and First Occurrence

In probability problems that involve competing patterns, the focus is on calculating which pattern is likely to appear first in a sequence of trials. This involves analyzing the structure and possible overlaps within patterns, and considering how one pattern might be prevented from forming if a competing pattern appears first.

Markov Chain Modeling in Pattern Occurrence

Markov chains are often used to model the progression of states leading to the occurrence of a particular pattern. By defining states that represent partial matches of the desired patterns, one can analyze transition probabilities between these states to determine the likelihood that one pattern emerges before another.

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