Binomial Probability Model and Bernoulli Trials

Definition of the Binomial Model: This model provides the probability for a discrete random variable that counts the total number of successes occurring within a fixed number of Bernoulli trials.
Model Parameters: The Binomial model is defined by two specific parameters: * $n$ : Represents the total number of trials performed. * $p$ : Represents the probability of success for an individual trial.
Notation: The model is formally denoted as $Binom(n, p)$ .
Comparison of Models: * Binomial: Focuses on the number of successes ( $k$ ) within a known batch or fixed number of trials ( $n$ ). * Geometric: Focuses on the number of trials required until the first success occurs ("play until success").

The Basis of Probabilistic Models: Bernoulli trials serve as the fundamental building blocks for the models examined in this chapter.
Requirements for Bernoulli Trials: To qualify as Bernoulli trials, the following three conditions must be met: 1. Two Possible Outcomes: Each trial must result in one of only two possible outcomes, classified strictly as "success" and "failure." 2. Constant Probability: The probability of attaining a success, denoted as $p$ , must remain constant across all trials. 3. Independence: Every trial must be independent of every other trial.

The Independence Challenge: One of the most critical requirements for Bernoulli trials is that they be independent. However, when sampling from a finite (non-infinite) population, trials are technically not independent because the outcome of one trial changes the probability for subsequent trials.
The 10% Condition Rule: There is a specific rule that allows statisticians to proceed by assuming independence even when it is strictly violated. The rule states that it is acceptable to proceed with the Bernoulli trial assumptions as long as the sample size is smaller than $10\%$ of the total population.
Notation for the 10% Rule: If the sample size is $n$ and the population is $N$ , the condition is often expressed as n < 0.10 \times N.

Model Variables and Definitions: * $Binom(n, p)$ : The identifier for the specific Binomial model being used. * $n$ : The number of trials, often described as the "handful" of events being looked at. * $p$ : The probability of a successful outcome. * $q$ : The probability of a failure, calculated as $q = 1 - p$ . * $X$ : The random variable representing the number of successes achieved in $n$ trials.
Expected Number of Successes (Mean): To find the expected number of successes ( $\mu$ ) in a batch of trials, use the following formula: $\mu = np$

Counting Successes: In a sequence of $n$ trials, there are multiple ways to achieve exactly $k$ successes. This is determined by the combinatorial formula.
Combination Formula: The number of ways to have $k$ successes is given by the combination formula: $C_n^k = \frac{n!}{k! \times (n-k)!}$
Terminology: The expression $C_n^k$ is read as "n choose k."
Factorials: The symbol "!" denotes a factorial. A factorial is calculated as the product of all positive integers from the number down to 1: $n! = n \times (n-1) \times \text{…} \times 2 \times 1$
Calculator Integration: Most modern graphing calculators have the "n choose k" function ( $\text{nCr}$ ) built-in to handle these calculations automatically.

The Probability Mass Function: The formula to calculate the probability of getting exactly $x$ successes in $n$ trials is: $P(X=x) = \binom{n}{x} p^x q^{n-x}$
Deconstruction of the Formula: * $\binom{n}{x}$ : Represents the number of different ways to arrange the successes and failures in the trials. * $p^x$ : Represents the probability of achieving the required number of successes. * $q^{n-x}$ : Represents the probability of achieving the remaining number of failures.
Implementation Note: Professional-grade or academic calculators typically handle these specific probability calculations ( $P(X=x)$ ) for the user, requiring only the input of the parameters $n$ , $p$ , and $x$ .