Binomial Probability Distributions – 5.2:Essentials of Statistics (Chapter 5 Study Notes)

Discrete Probability Distributions

Introduction to Discrete Probability Distributions

Discrete probability distributions describe the probabilities of outcomes for discrete random variables, which can take on a countable number of values. These distributions are fundamental in statistics for modeling scenarios where outcomes are distinct and separate.

• Discrete Probability Distribution: A listing or function that gives the probability of each possible value of a discrete random variable.

• Main Types: Binomial and Poisson probability distributions are commonly studied in introductory statistics.

• Applications: Used in quality control, genetics, reliability engineering, and many other fields.

Binomial Probability Distributions

Key Concept: Binomial Probability Distribution

The binomial probability distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It is widely used to answer questions about the likelihood of a certain number of successes in repeated experiments.

• Definition: The probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes/no question, and each with its own boolean-valued outcome.

• Examples:

  • Flipping a fair coin 10 times and counting the number of heads.

  • Transmitting bits through a digital channel and counting the number of errors.

  • Testing air samples for a rare molecule and counting the number of positive samples.

Requirements for a Binomial Probability Distribution

For a random experiment to be modeled by a binomial distribution, it must satisfy four key requirements:

• Fixed Number of Trials: The procedure has a predetermined number of trials, denoted by n.

• Independence: Each trial is independent; the outcome of one trial does not affect the others.

• Two Categories: Each trial results in one of exactly two possible outcomes, commonly called success and failure.

• Constant Probability: The probability of success, denoted by p, remains the same for each trial.

Notation and Parameters

Standard notation is used to describe the binomial distribution:

• n: Number of trials

• x: Number of successes (can be any integer from 0 to n)

• p: Probability of success in a single trial

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

Note: The term "success" is arbitrary and does not necessarily mean something positive; it simply refers to the outcome being counted.

Binomial Probability Formula

The probability of getting exactly x successes in n independent trials is given by the binomial formula:

• Formula:

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

• Where:

  • $n!$ is the factorial of n

  • $p$ is the probability of success

  • $q$ is the probability of failure ($q = 1 - p$)

  • $x$ is the number of successes

Example: Cashless Adults

Suppose the probability that a randomly selected adult smartphone owner is cashless is 0.05. What is the probability that, among 10 randomly selected adults, exactly 2 are cashless?

• Parameters: $n = 10$, $x = 2$, $p = 0.05$, $q = 0.95$

• Calculation:

$P(2) = \frac{10!}{2!8!} (0.05)^2 (0.95)^8$

Evaluating this gives $P(2) \approx 0.0746$ (rounded to three significant digits).

Example: Vegetarians

Based on a poll, 5% of U.S. adults are vegetarians. If five adults are randomly selected, find:

• Probability that exactly two are vegetarians:

$P(2) = \frac{5!}{2!3!} (0.05)^2 (0.95)^3 = 10 \times 0.0025 \times 0.857375 \approx 0.021$

• Probability that fewer than three are vegetarians:

$P(0 \text{ or } 1 \text{ or } 2) = P(0) + P(1) + P(2) = 0.774 + 0.204 + 0.021 = 0.999$

Mean and Standard Deviation of Binomial Distributions

Formulas for Mean, Variance, and Standard Deviation

The mean, variance, and standard deviation of a binomial distribution are given by:

• Mean: $\mu = np$

• Variance: $\sigma^2 = npq$

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

These formulas allow us to describe the expected value and spread of the distribution.

Range Rule of Thumb for Significance

The range rule of thumb helps identify values that are significantly low or high in a binomial distribution:

• Significantly low values: Less than $\mu - 2\sigma$

• Significantly high values: Greater than $\mu + 2\sigma$

• Not significant: Between $\mu - 2\sigma$ and $\mu + 2\sigma$

Example: NFL Overtime Wins

Suppose in 460 NFL overtime games, the probability of winning is 0.5 for each team. Find the mean and standard deviation for the number of wins, and determine if 252 wins is significantly high.

• Parameters: $n = 460$, $p = 0.5$, $q = 0.5$

• Mean: $\mu = 460 \times 0.5 = 230.0$

• Standard Deviation: $\sigma = \sqrt{460 \times 0.5 \times 0.5} = \sqrt{115} \approx 10.7$

• Significantly high threshold: $\mu + 2\sigma = 230.0 + 2 \times 10.7 = 251.4$

• Conclusion: 252 wins is significantly high because it exceeds 251.4.

Summary Table: Binomial Distribution Properties

Parameter	Symbol	Formula	Description
Number of trials	n	-	Fixed number of independent experiments
Probability of success	p	-	Probability of success in a single trial
Probability of failure	q	$q = 1 - p$	Probability of failure in a single trial
Mean	$\mu$	$\mu = np$	Expected number of successes
Variance	$\sigma^2$	$\sigma^2 = npq$	Measure of spread
Standard Deviation	$\sigma$	$\sigma = \sqrt{npq}$	Square root of variance
Probability of x successes	$P(x)$	$P(x) = \frac{n!}{x!(n-x)!} p^x q^{n-x}$	Probability of exactly x successes in n trials