THE BINOMIAL AND NORMAL
PROBABILITY MODELS
CHAPTER 4
Comprehensive Exam Study Guide
Latest Updated 2024/2025
TABLE OF CONTENTS
Page
1 The Binomial Probability Distribution 112
Conditions for a Binomial Experiment, Bernoulli Trials 112
Mean and Standard Deviation of a Binomial Random Variable 112
Using Excel and Table 1 to Calculate Binomial Probabilities 113
2 The Normal Probability Model 119
The Standard Normal Distribution 119
Standardizing Normal Distributions 125
Mensa Example 126
Tele-Evangelist Example 127
3 Approximating a Binomial Model with the Normal Distribution 127
Advertising Campaign and Dear Abby Examples 128
ESP Example 129
4 Fundamental Ideas From The Fourth Session 130
Six Probability Model Exercises 131
Answer Sketches for Six Probability Model Exercises 133
5 Table One: The Binomial Distribution 135
6 Table Two: The Normal Distribution 143
, THE BINOMIAL AND NORMAL
PROBABILITY MODELS
CHAPTER 4
Comprehensive Exam Study Guide
Latest Updated 2024/2025
The Binomial Probability Distribution
The simplest data-gathering process is counting the number of times a certain event occurs. We can
reduce almost an endless variety of situations to this simple counting process.
Consumer preference and opinion polls (i.e., sample surveys) are conducted frequently in business.
Some recent examples include polls to determine the number of voters in favor of legalizing casino
gambling in a particular state, and the number of Americans in favor of government-controlled health
care.
The Binomial distribution is a discrete probability distribution that is extremely useful for describing many
phenomena. The random variable of interest that follows the Binomial distribution is the number of
successes obtained in a sample of n observations. The Binomial distribution can be applied to
numerous applications, such as:
What is the probability that in a sample of 20 tires of the same type none will be defective if 10% of
all such tires produced at a particular plant are defective?
What is the probability that red will come up 15 or more times in 20 spins of the American roulette
wheel (38 spaces)?
What is the probability that a student can pass (that is, get at least 60% correct on) a 30-question
multiple-choice exam (each question containing four choices) if the student guesses each question?
What is the probability that a particular stock will show an increase in its closing price on a daily
basis over the next ten (consecutive) trading sessions if stock market price changes really are
random?
Conditions Required for a Binomial Experiment
1. There is a set of n trials, which can be classified as either “successes” or “failures.”
2. The probability of a success, , is constant over the n trials.
3. The outcome for any one trial is independent of the outcome for any other trial.
These first three conditions specify Bernoulli Trials.
4. The Binomial random variable X counts the number of “successes” in n trials.
, THE BINOMIAL AND NORMAL
PROBABILITY MODELS
CHAPTER 4
Comprehensive Exam Study Guide
Latest Updated 2024/2025
Mean and Standard Deviation of the Binomial Probability Distribution
A random variable following the Binomial distribution is completely specified by the two parameters n
, THE BINOMIAL AND NORMAL
PROBABILITY MODELS
CHAPTER 4
Comprehensive Exam Study Guide
Latest Updated 2024/2025
and . Note that for a random variable X following the Binomial (n, ) distribution:
m X E X np , and s X np 1 p
We can obtain binomial probabilities using a Table (like Table 1) or Excel.
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