Consider a sequence of independent Bernoulli trials with...

Consider a sequence of independent Bernoulli trials with $p=0.2$.
(a) What is the expected number of trials to obtain the first success?
(b) After the eighth success occurs, what is the expected number of trials to obtain the ninth success?

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Key Concepts

Bernoulli Trials

Bernoulli trials are experiments or processes that result in one of two outcomes, success or failure, with the probability of success remaining constant across trials and each trial being independent of the others. This concept is foundational for modeling binary events in probability theory.

Geometric Distribution

The geometric distribution models the number of independent Bernoulli trials needed to achieve the first success. It shows that the expected number of trials is the inverse of the probability of success, demonstrating a direct relationship between success probability and trial count.

Negative Binomial Distribution

The negative binomial distribution generalizes the geometric distribution by modeling the number of trials needed to achieve a specified number of successes. It is particularly useful when dealing with scenarios where multiple successes are required, and it provides the expected number of additional trials needed after a certain count of successes.

Memorylessness

Memorylessness is a property inherent in the geometric distribution whereby the probability of success in future trials does not depend on the number of past failures. This property simplifies analysis by ensuring that the process ‘restarts’ after each trial regardless of the history.

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