(a) Independent Bernoulli trials are performed, with...

(a) Independent Bernoulli trials are performed, with probability $1 / 2$ of success, until there has been at least one success. Find the PMF of the number of trials performed. (b) Independent Bernoulli trials are performed, with probability $1 / 2$ of success, until there has been at least one success and at least one failure. Find the PMF of the number of trials performed.

(a) Independent Bernoulli trials are performed, with probability $1 / 2$ of success, until there has been at least one success. Find the PMF of the number of trials performed.
(b) Independent Bernoulli trials are performed, with probability $1 / 2$ of success, until there has been at least one success and at least one failure. Find the PMF of the number of trials performed.

Key Concepts

Bernoulli Trials

Bernoulli trials are experiments or processes where each trial results in one of two outcomes, typically labeled success and failure, with a fixed probability for success in each trial. They form the foundational building block for many discrete probability distributions and are assumed to be independent of one another in many probabilistic models.

Geometric Distribution

The geometric distribution models the number of independent Bernoulli trials needed to achieve the first success, assuming each trial is independent and has the same probability of success. This distribution is characterized by its memoryless property, meaning that the probability of success in future trials does not depend on the number of failures that have already occurred.

Probability Mass Function (PMF)

The probability mass function (PMF) is a function that gives the probability of a discrete random variable taking on a particular value. It is a fundamental concept in probability theory, as it fully describes the distribution of a discrete random variable by assigning probabilities to each possible outcome.

Stopping Time

A stopping time is a random variable that represents the time at which a given stochastic process meets a specific condition or criterion for the first time. In the context of Bernoulli trials, stopping times can be used to define when a sequence should be terminated based on outcomes, such as stopping after the first success or after observing both a success and a failure.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. This concept is essential when the outcome of a process depends on previous outcomes, such as in sequential trials where the probability of future events may need to be calculated based on the results of initial trials.

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