5.2 The Binomial_Distribution

Binomial Distribution

Definition: A Binomial Probability Distribution arises from trials with two outcomes: success or failure.

Examples: Defective items or types of clothing.

Requirements:

• Fixed number of trials (n).

• Two possible outcomes (success/failure).

• Constant probability of success (p).

• Independent trials.

• Random variable x denotes successes in n trials.

Notation:

• n = number of trials

• p = probability of success

• q = probability of failure (p + q = 1)

• x = number of successes in n trials

• X = {0, 1, …, n} (discrete random variable)

Binomial Probability Formula:

• P(x=k) = (\binom{n}{k}) p^k q^{n-k}, where \binom{n}{k} is the number of ways to achieve k successes.

Parameters:

• Expected value (mean): E(X) = n * p

• Standard deviation: \sigma_x = \sqrt{n * p * q}

Examples:

1. Surprise Quiz: 4 questions, p = 1/5.

   • Probability Distribution Table given for number of correct answers.

2. Americans Afraid to Stay Home: 7% afraid (n=25).

   • P(exactly 5 afraid) = binomialpdf(25, 0.07, 5) = 0.0209

   • Expected value: E(X) = 1.75, standard deviation = 1.3.

Examples of Applying Binomial Distribution Concepts

1. Quality Control in Manufacturing:A company produces electronic components and wants to determine the probability of finding defective items in a batch. If the probability of a component being defective is 0.1 (p = 0.1) and they test 20 components (n = 20), they can use the binomial distribution to calculate the likelihood of finding a certain number of defects.

2. Marketing Research:A marketing team wants to analyze customer feedback after launching a new product. If they survey 100 people and find that 30% like the product (p = 0.3), they can model the number of people who express satisfaction using the binomial distribution.

3. Medical Studies:In a clinical trial, researchers want to see how many patients respond positively to a treatment. If they have 50 patients (n = 50) and know that historically, 60% (p = 0.6) of patients respond positively, they can calculate the probabilities of various outcomes for treatment success.

4. Polls and Surveys:If a pollster wants to find out how many people support a particular policy in a community of 1000 adults and estimates that 40% (p = 0.4) of them will support it, they can use the binomial distribution to determine the likelihood of different numbers of supporters.