State the criteria for a binomial probability experiment

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What is a binomial probability experiment?

A binomial probability experiment is a statistical experiment that must satisfy four specific conditions simultaneously. The experiment involves conducting a fixed number of identical trials where each trial produces one of two possible outcomes and maintains independence from other trials.

What are the four criteria for a binomial probability experiment?

A binomial probability experiment requires four criteria. Each criterion must be present for an experiment to qualify as binomial.

1. Fixed number of trials

The experiment contains a predetermined number of trials, denoted by n. The number of trials remains constant throughout the experiment and is established before conducting any observations.

2. Independence of trials

Each trial operates independently of all other trials. The outcome of one trial does not influence or affect the probability of outcomes in subsequent trials. The trials maintain complete independence throughout the experiment.

3. Two possible outcomes per trial

Each trial produces exactly two mutually exclusive outcomes. These outcomes are labeled as "success" and "failure" for analytical purposes, regardless of the positive or negative connotation of the actual results.

4. Constant probability of success

The probability of success, denoted by p, remains identical across all trials. The probability does not change from one trial to another throughout the experiment. The probability of failure, denoted by q, equals 1-p and remains constant as well.

What notation represents binomial probability experiments?

Binomial probability experiments use standardized mathematical notation. The number of trials is denoted by n. The probability of success in each trial is denoted by p. The number of successes observed is represented by X as a random variable. The probability of failure in each trial is denoted by q, where q=1-p. The binomial distribution of the random variable X with parameters n and p is written as:

$X\sim B(n,p)$

What are examples of binomial probability experiments?

Coin flipping demonstrates a binomial experiment where each flip results in heads (success) or tails (failure). Quality control testing uses binomial experiments when inspecting a fixed number of products to determine how many are defective. Medical trials apply binomial models when testing a fixed number of patients to see how many respond positively to a treatment.

Additional examples include counting the number of side effects from medications in a fixed patient group, fraud detection in bank transactions modeling the number of fraudulent transactions out of many daily transactions, counting spam emails received per day in an account, and retail scenarios measuring the number of returned items out of total sales in a given period.

These examples satisfy all four binomial criteria: fixed number of trials, independence between trials, two possible outcomes per trial, and constant success probability.

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