4.2- Binomial Distributions: Concepts, Formulas, and Applications

Section 4.2: Binomial Distributions

Introduction to Binomial Distributions

The binomial distribution is a fundamental probability distribution in statistics, used to model the number of successes in a fixed number of independent trials, each with the same probability of success. This section covers how to identify binomial experiments, calculate binomial probabilities, and determine the mean, variance, and standard deviation of a binomial distribution.

Binomial Experiments

Definition and Characteristics

• Binomial Experiment: An experiment that meets all the following criteria:

  • Consists of a fixed number of trials, denoted by n.

  • Each trial is independent of the others.

  • Each trial has only two possible outcomes: success or failure.

  • The probability of success, p, is the same for each trial.

  • The random variable x counts the number of successes in n trials.

• Examples:

  • Flipping a coin (heads or tails).

  • Determining the gender of a newborn (boy or girl).

Notation

• n: Number of trials

• x: Number of successes

• p: Probability of success in a single trial

• q: Probability of failure in a single trial ($q = 1 - p$)

Success vs. Failure

Understanding Success and Failure

• The terms "success" and "failure" are arbitrary and do not necessarily represent good or bad outcomes.

• The definition of "success" is determined by the researcher's question.

• It is important to ensure that both outcomes are clearly defined and mutually exclusive.

Identifying Binomial Experiments

Examples

• Example 1: A medical procedure with an 85% chance of success. If 8 patients undergo the procedure, the number of successes is a binomial random variable.

• Example 2: Drawing marbles from a bag with replacement. The number of red marbles drawn in 5 trials is a binomial random variable.

• Example 3: Multiple-choice test with 10 questions, each with 4 options. The number of correct answers by random guessing is a binomial random variable.

Binomial Probability Formula

Calculating Binomial Probabilities

In a binomial experiment, the probability of exactly x successes in n trials is given by:

• $P(x) = \binom{n}{x} \cdot p^x \cdot q^{n-x}$

• Where $\binom{n}{x} = \frac{n!}{x!(n-x)!}$ is the binomial coefficient.

• n! denotes the factorial of n.

Example: For n = 4, x = 1: $\binom{4}{1} = 4$

Using Calculators and Tables

• Most scientific calculators have a function for binomial coefficients (often labeled as nCr).

• Binomial distribution tables can be used to look up probabilities for given values of n, x, and p.

Binomial Distribution Table

Purpose and Usage

• Tables provide pre-calculated probabilities for various combinations of n, x, and p.

• To use the table, locate the row for the number of successes (x) and the column for the probability of success (p).

• Tables are especially useful for large values of n or when calculations are complex.

Worked Examples

Example 4

• Blood donors: n = 16, x = 6, p = 0.45, q = 0.55

• $P(x = 6) = \binom{16}{6} \cdot 0.45^6 \cdot 0.55^{10} = 0.168$

Example 5

• Burglar alarm system with 5 fail-safe components, each with a 5% probability of failure. Find the probability that at most 2 will fail.

• $P(x \leq 2) = P(x = 0) + P(x = 1) + P(x = 2) = 0.735 + 0.232 + 0.031 = 0.998$

Example 6

• TV selection: n = 10, x = 9 or 10, p = 0.80, q = 0.20

• $P(x = 9 \text{ or } 10) = P(x = 9) + P(x = 10) = 0.268 + 0.107 = 0.375$

Example 7

• True/False exam: n = 15, p = 0.5, find probability of passing by guessing (more than 10 correct answers).

• $P(x = 11, 12, 13, 14, 15) = 0.042 + 0.014 + 0.003 + 0.000 + 0.000 = 0.059$

Mean, Variance, and Standard Deviation of Binomial Distribution

Formulas

• Mean: $\mu = np$

• Variance: $\sigma^2 = npq$

• Standard Deviation: $\sigma = \sqrt{npq}$

Example 8

• MM&M candy: n = 104, p = 0.126

• Mean: $\mu = 104 \times 0.126 = 13.104$

• Variance: $\sigma^2 = 104 \times 0.126 \times 0.874 = 11.45$

• Standard Deviation: $\sigma = \sqrt{11.45} = 3.38$

Example 9

• Federal government employees using e-mail: n = 200, p = 0.83

• Mean: $\mu = 200 \times 0.83 = 166$

• Variance: $\sigma^2 = 200 \times 0.83 \times 0.17 = 28.22$

• Standard Deviation: $\sigma = \sqrt{28.22} = 5.31$

Summary Table: Binomial Distribution Properties

Parameter	Symbol	Formula	Description
Mean	$\mu$	$np$	Expected number of successes
Variance	$\sigma^2$	$npq$	Measure of spread
Standard Deviation	$\sigma$	$\sqrt{npq}$	Square root of variance

Additional info:

• Binomial tables are typically found in the appendix of statistics textbooks and are used for quick probability lookups.

• Calculator instructions may vary by model; refer to your calculator's manual for binomial functions.