Chapter 16 - Geometric and Binomial Probability Models

Foundational Context: The study of Geometric and Binomial models falls under the broader category of "Modeling the World."
The Role of Conditions: Statistical models are not universal; they are built upon specific sets of conditions or assumptions. If these conditions are not met, the model may not be appropriate for the data. * Normal Model: Requires that data be unimodal and symmetric. * Linear Model: Requires that the data exhibits a linear relationship. * Geometric and Binomial Models: These models are built upon the foundation of Bernoulli trials.

Definition: The Bernoulli trial is the fundamental basis for the probability models examined in this chapter.
Criteria for Bernoulli Trials: To qualify as Bernoulli trials, the following three conditions must be satisfied: 1. Two Possible Outcomes: There must be exactly two possible outcomes for each trial, categorized as "success" and "failure." 2. Constant Probability ( $p$ ): The probability of success, denoted by $p$ , must remain constant across all trials. Consequently, the probability of failure ( $q$ ) is also constant ( $q = 1 - p$ ). 3. Independent Trials: The outcome of one trial must not influence or change the outcome or probability of any other trial.

The Independence Requirement: A central assumption for Bernoulli trials is that the trials are independent of one another.
Challenges with Finite Populations: In practice, when sampling from a population that is not infinite, removing individuals for the sample changes the probability for subsequent trials, meaning the trials are technically not independent.
The Rule of Thumb (The 10% Condition): * Specific Rule: It is acceptable to "pretend" that trials are independent and proceed with Bernoulli-based models even if the true independence assumption is technically violated, provided the sample size is small. * The Threshold: The sample must be smaller than $10\%$ of the total population. * Implication: Keeping the sample under $10\%$ ensures that the probability $p$ remains approximately constant enough that the trials can be treated as independent for calculation purposes.

Model Designation: Represented as $Geom(p)$ , where $p$ is the probability of success.
Core Objective: The Geometric model is used to find the probability of the number of trials required until the first success occurs.
Variable Definitions: * $p$ : The probability of a success. * $q$ : The probability of a failure, where $q = 1 - p$ . * $X$ : The random variable representing the number of trials until the first success occurs.
The "Failure Until Final Success" Logic: The model describes a sequence of events where a series of failures occurs, concluding with a single, final success.
Probability Mass Function (PMF): * The probability that the first success occurs on the $x$ -th trial is calculated as: $P(X = x) = q^{x-1} p$ * This can also be written substituting for $q$ : $P(X = x) = (1 - p)^{x-1} p$ * Conceptual Breakdown: The formula signifies $(\text{Probability of Failure})^{x-1} \times (\text{Probability of Success})^1$ .

Mean ( $\mu$ ): * Also known as the Expected Value ( $E(x)$ ). * Formula: $\mu = E(x) = \frac{1}{p}$ . * Definition: This represents the expected number of trials or "picks" required until you achieve your first success.
Standard Deviation ( $\sigma$ ): * Formula: $\sigma = \frac{\sqrt{q}}{p}$ .