Binomial Probability Distributions: Concepts, Notation, and Applications

Probability Distributions

Introduction

Probability distributions are fundamental in statistics for describing the likelihood of different outcomes in random experiments. This section focuses on binomial probability distributions, including their requirements, notation, calculation methods, and interpretation of results.

Binomial Probability Distribution

Definition and Requirements

A binomial probability distribution arises from a procedure that meets four specific requirements:

• Fixed number of trials: The experiment consists of a predetermined number of trials, denoted by n. Each trial is a single observation.

• Independence: The outcome of any individual trial does not affect the probabilities in other trials.

• Two categories: Each trial results in one of two possible outcomes, commonly referred to as success and failure.

• Constant probability of success: The probability of success, denoted by p, remains the same for all trials.

These requirements ensure that the binomial model is appropriate for the scenario.

Notation for Binomial Probability Distributions

• S and F: Denote the two possible categories of outcomes (success and failure).

• P(S) = p: Probability of a success in one trial.

• P(F) = q = 1 - p: Probability of a failure in one trial.

• n: Fixed number of trials.

• x: Specific number of successes in n trials (where x can be any integer from 0 to n).

• P(x): Probability of getting exactly x successes among the n trials.

Important Note: The term "success" is arbitrary and does not necessarily represent something positive. It is essential that x and p consistently refer to the same category being called a success.

Example: Cashless Smartphone Owners

Problem Statement

Suppose the probability that a randomly selected adult smartphone owner is cashless is 0.05. If ten adults are randomly selected, what is the probability that exactly two of them are cashless?

• Does this procedure result in a binomial distribution? Yes, because:

  • The number of trials (n = 10) is fixed.

  • Trials are independent.

  • Each trial has two outcomes: cashless or not.

  • The probability of being cashless (p = 0.05) is constant.

• Identify the values:

  • n = 10

  • x = 2

  • p = 0.05

  • q = 0.95

Methods for Finding Binomial Probabilities

Method 1: Binomial Probability Formula

The probability of getting exactly x successes in n trials is given by:

$P(x) = \frac{n!}{(n-x)!\,x!} \cdot p^x \cdot q^{n-x}$ for $x = 0, 1, 2, \ldots, n$

Where:

• n: Number of trials

• x: Number of successes

• p: Probability of success

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

Example Calculation

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

$P(2) = \frac{10!}{8!\,2!} \cdot 0.05^2 \cdot 0.95^{8}$ $= 45 \cdot 0.0025 \cdot 0.663420 = 0.074635$ $\approx 0.0746$ (rounded to three significant digits)

Interpretation: The probability of getting exactly two cashless adults is 0.0746.

Method 2: Using Excel

Excel provides built-in functions to calculate binomial probabilities efficiently. The function BINOM.DIST can be used as follows:

Syntax: =BINOM.DIST(x, n, p, FALSE)

For example, =BINOM.DIST(2, 10, 0.05, FALSE) calculates the probability of exactly two successes in ten trials with a success probability of 0.05.

x	P(x)
0	0.5987369
1	0.3150935
2	0.0746350
3	0.0107527
4	0.0009226
5	0.0000490
6	0.0000017
7	0.0000000
8	0.0000000
9	0.0000000
10	9.765625e-14

Method 3: Using Table A-1 in Appendix A

Table A-1 lists binomial probabilities for select values of n and p. It is useful for small values of n (typically $n \leq 8$) and specific probabilities. To use the table:

• Locate the row for the desired n and x.

• Find the column for the desired p.

• The intersection gives $P(x)$.

Example: For $n = 5$, $p = 0.05$, $x = 2$, $P(2) = 0.021$.

Interpreting Binomial Probabilities

Significantly High and Low Values

Statistical significance is determined by comparing observed values to expected values using probability and the range rule of thumb.

• Significantly high: $x$ is significantly high if $P(x \text{ or more}) < 0.05$.

• Significantly low: $x$ is significantly low if $P(x \text{ or fewer}) < 0.05$.

Range Rule of Thumb

• Significantly low values: $\leq (\mu - 2\sigma)$

• Significantly high values: $\geq (\mu + 2\sigma)$

• Values not significant: Between $(\mu - 2\sigma)$ and $(\mu + 2\sigma)$

Mean and Standard Deviation of Binomial Distributions

Formulas

• Mean: $\mu = np$

• Variance: $\sigma^2 = npq$

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

Example: Overtime Rule in Football

Suppose $n = 460$ games, $p = 0.5$, $q = 0.5$.

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

• Standard deviation: $\sigma = \sqrt{460 \times 0.5 \times 0.5} = 10.7$ games

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

• Significantly low: $\mu - 2\sigma = 230.0 - 2 \times 10.7 = 208.6$ games

Thus, 252 wins is considered significantly high, as it exceeds 251.4.

Treating Dependent Events as Independent

5% Guideline for Cumbersome Calculations

When sampling without replacement, if the sample size is no more than 5% of the population size, selections can be treated as independent for practical purposes, even though they are technically dependent.

Summary Table: Binomial Probability Distribution Notation

Symbol	Meaning
n	Number of trials
x	Number of successes
p	Probability of success in one trial
q	Probability of failure in one trial ($q = 1 - p$)
P(x)	Probability of exactly x successes in n trials

Additional info:

• Binomial probability distributions are widely used in quality control, genetics, and survey analysis.

• Excel and other statistical software can efficiently compute binomial probabilities for large sample sizes.