Binomial Distributions and Discrete Probability Distributions: Study Notes for Introductory Statistics

Discrete Probability Distributions

Binomial Distributions

Binomial distributions are a fundamental topic in introductory statistics, describing the probability of a fixed number of successes in a series of independent trials, each with the same probability of success. Understanding binomial distributions is essential for analyzing experiments with two possible outcomes.

• Binomial Experiment Criteria:

  • The experiment is repeated for a fixed number of trials (n).

  • Each trial is independent of the others.

  • Each trial has only two possible outcomes: success (S) or failure (F).

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

  • The random variable x counts the number of successful trials.

• Notation:

  • n: Number of trials

  • p: Probability of success

  • q: Probability of failure (q = 1 - p)

  • x: Number of successes in n trials

Identifying Binomial Experiments

To determine if an experiment is binomial, check if it meets all the criteria above. If the probability of success changes between trials or the trials are not independent, it is not a binomial experiment.

• Example: A doctor performs a surgical procedure with an 85% success rate on eight patients. Each surgery is independent, and the probability of success is constant. This is a binomial experiment with n = 8, p = 0.85, q = 0.15, and x = 0, 1, ..., 8.

• Non-Binomial Example: Drawing marbles from a jar without replacement changes the probability of success for each trial, making the trials dependent. This is not a binomial experiment.

Binomial Probability Formula

The binomial probability formula calculates the probability of exactly x successes in n trials:

$P(x) = \frac{n!}{x!(n-x)!} p^x q^{n-x}$

• n: Number of trials

• x: Number of successes

• p: Probability of success

• q: Probability of failure (q = 1 - p)

Finding Binomial Probabilities

Binomial probabilities can be found using:

• Tree diagrams and the Multiplication Rule

• The binomial probability formula

• Technology (e.g., Minitab, Excel, StatCrunch, TI-84 Plus)

• Binomial probability tables

Example: Using the Binomial Formula

Suppose rotator cuff surgery has a 90% success rate and is performed on three patients. The probability that exactly two surgeries are successful is:

$P(2) = \frac{3!}{2!1!} (0.9)^2 (0.1)^1 = 3 \times 0.81 \times 0.1 = 0.243$

Example: Using Technology

For large n, technology is often used. For example, if 75% of adults experience digital device fatigue and you select 100 adults, the probability that exactly 66 experience fatigue can be found using statistical software, yielding approximately 0.011.

Example: Using Binomial Tables

Binomial tables provide probabilities for specific values of n, p, and x. For example, with n = 8, p = 0.05, and x = 3, the probability is 0.005.

Constructing and Graphing Binomial Distributions

A binomial probability distribution lists the probabilities for each possible value of x. The sum of all probabilities in the distribution is 1. Graphs such as histograms can visually represent the distribution.

• Example: If 64% of cancer survivors are 65 or older, and you select six survivors, the distribution for x (number aged 65+) can be graphed. Probabilities for x = 0 or 1 are less than 0.05, indicating these are unusual outcomes.

Mean, Variance, and Standard Deviation of Binomial Distributions

The mean, variance, and standard deviation of a binomial distribution are calculated as follows:

• Mean: $\mu = n p$

• Variance: $\sigma^2 = n p q$

• Standard Deviation: $\sigma = \sqrt{n p q}$

Example: Cloudy Days in June

In Pittsburgh, 56% of days are cloudy. For June (n = 30), the mean number of cloudy days is:

$\mu = 30 \times 0.56 = 16.8$

Variance:

$\sigma^2 = 30 \times 0.56 \times 0.44 = 7.392$

Standard deviation:

$\sigma = \sqrt{7.392} \approx 2.7$

Values more than two standard deviations from the mean are considered unusual (e.g., 11 or fewer, or 23 or more cloudy days).

Summary Table: Binomial Distribution Properties

Property	Formula	Description
Mean	$\mu = n p$	Expected number of successes
Variance	$\sigma^2 = n p q$	Spread of the distribution
Standard Deviation	$\sigma = \sqrt{n p q}$	Typical deviation from the mean

Applications and Interpretation

• Binomial distributions are used in medical studies, surveys, and quality control.

• Unusual events are those with probabilities less than 0.05.

• Technology and tables simplify calculations for large n.

Additional info: The included image is the cover of the textbook "Elementary Statistics" by Ron Larson, which is directly relevant as it visually identifies the source and context of the study notes.