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Discrete Probability Distributions

Chapter 06

McGraw-Hill/Irwin

Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved.

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The study of probability begins with understanding the two basic classifications of a random variable–discrete and continuous. In order to apply the correct formula for solving a probability question, one must be able to know whether the variable is discrete or continuous.

LEARNING OBJECTIVES

LO 6-1 Identify the characteristics of a probability distribution.

LO 6-2 Distinguish between a discrete and a continuous random variable.

LO 6-3 Compute the mean of a probability distribution.

LO 6-4 Compute the variance and standard deviation of a probability distribution.

LO 6-5 Describe and compute probabilities for a binomial distribution.

LO 6-6 Describe and compute probabilities for a Poisson distribution.

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In this chapter, we investigate the commonly occurring discrete probability distributions–namely, the uniform, binomial, and poisson. We explore examples demonstrating situations involving these discrete distributions.

 

We are going to define the terms probability distribution and random variable. We will learn the distinction between discrete and continuous probability distributions. For a discrete probability distribution, we will learn how to calculate its mean, variance, and standard deviation. We’re also going to describe the characteristics of and compute probabilities using the binomial and the Poisson probability distributions.

What Is a Probability Distribution?

Experiment:

Toss a coin three times. Observe the number of times heads appears.

The possible results are:

Zero heads

One heads

Two heads

Three heads

What is the probability distribution for the number of heads?

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PROBABILITY DISTRIBUTION A listing of all the outcomes of an experiment and the probability associated with each outcome.

LO 6-1 Identify the characteristics of a probability distribution.

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A probability distribution is a listing of all the outcomes of an experiment and the probability associated with each outcome. Probability distributions are characterized by the following: the probability of a particular outcome, usually denoted by P of x, is between zero and one inclusive. The outcomes, denoted by x, are mutually exclusive; and the list is exhaustive so that the sum of the probabilities of the various outcomes is equal to 1.

 

For example, consider an experiment involving tossing a coin three times. We let x represent the number of heads observed in three tosses. The exhaustive list for x list is zero heads, one heads, two heads, and three heads. Altogether, there are eight possible unique combinations of how the three coins will appear, as shown in the table in the slide. The probability of observing zero heads is one of eight outcomes. Since the list of x is exhaustive, the sum of their probabilities is 1.

Characteristics of a Probability Distribution

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  • The probability of a particular outcome is between 0 and 1 inclusive.

2. The outcomes are mutually exclusive events.

3. If the list is collectively exhaustive, the sum of the probabilities of the various events is 1.

LO 6-1

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Discrete probability distributions have the following characteristics:

One, the probability of a particular outcome is between 0 and 1 inclusive.

Two, the outcomes are mutually exclusive events.

Three, the list is exhaustive. So the sum of the probabilities of the various events is equal to 1.

Probability Distribution of the Number of Heads Observed in 3 Tosses of a Coin

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LO 6-1

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In an earlier slide, we considered an experiment involving tossing a coin three times. We were interested in observing the number of heads in three tosses. We denoted this random number as x. The table on the slide shows the exhaustive list for x : zero heads, one heads, two heads, and three heads. Altogether, there are eight possible unique combinations of how the three coins will appear, shown in the table in the slide. The probability of observing zero heads is one of eight outcomes, or 0.125. Because x is listed exhaustively, the sum of their probabilities is equal to one. The figure on the right of the table graphs the probabilities of the respective outcomes. The resulting graph outlines the shape of the probability distribution of this coin tossing experiment.

Random Variables

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RANDOM VARIABLE A quantity resulting from an experiment that, by chance, can assume different values.

LO 6-1

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We define random variable. A random variable is a value or number that results from an experiment that by chance can take on different values. We usually use the letter x to denote a random variable.

 

In the coin tossing experiment discussed in the previous slide, the random variable of the experiment is the number of heads that will appear in three tosses of a coin. The possible values are 0, 1, 2, or 3.

Types of Random Variables

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DISCRETE RANDOM VARIABLE A random variable that can assume only certain clearly separated values. It is usually the result of counting something.

CONTINUOUS RANDOM VARIABLE Can assume an infinite number of values within a given range. It is usually the result of some type of measurement.

LO 6-2 Distinguish between a discrete and a continuous random variable.

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Earlier, we defined a random variable as an observation, a quantity, or a number resulting from an experiment that by chance can assume different values. Random variables are classified as either discrete or continuous. A random number that can assume only certain clearly separated values, usually the result of counting something, is known as a discrete random number.

 

Random numbers that can assume an infinite number of values within a range, usually the result of some type of measurement, are known as continuous random numbers.

Discrete Random Variables

EXAMPLES

The number of students in a class.

The number of children in a family.

The number of cars entering a carwash in an hour.

Number of home mortgages approved by Coastal Federal Bank last week.

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DISCRETE RANDOM VARIABLE A random variable that can assume only certain clearly separated values. It is usually the result of counting something.

LO 6-2

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Continuous Random Variables

EXAMPLES

  • The length of each song on the latest Tim McGraw album.
  • The weight of each student in this class.
  • The temperature outside as you are reading this book.
  • The amount of money earned by each of the more than 750 players currently on Major League Baseball team rosters.

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CONTINUOUS RANDOM VARIABLE Can assume an infinite number of values within a given range. It is usually the result of some type of measurement.

LO 6-2

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The Mean of a Probability Distribution

MEAN

  • A typical value used to represent the central location of a probability distribution.
  • The mean of a probability distribution is also referred to as its expected value.

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LO 6-3 Compute the mean of a probability distribution.

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Mean, Variance, and Standard
Deviation of a Probability Distribution – Example

John Ragsdale sells new cars for Pelican Ford. He has developed the following probability distribution for the number of cars he expects to sell on a particular Saturday.

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LO 6-3

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Mean of a Probability Distribution –
Example

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LO3

LO 6-3

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The Variance and Standard
Deviation of a Probability Distribution

  • Measure the amount of spread in a distribution
  • Computational steps:

1. Compute the mean

2. Subtract the mean from each value, and square this difference.

3. Multiply each squared difference by its probability.

4. Sum the resulting products to arrive at the variance.

5. Take the positive square root of the variance to obtain the standard deviation.

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LO 6-4 Compute the variance and standard deviation of a probability distribution.

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Variance and Standard
Deviation of a Probability Distribution – Example

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LO 6-4

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Binomial Probability Distribution

  • A widely occurring discrete probability distribution
  • Characteristics of a binomial probability distribution

There are only two possible outcomes of a particular trial of an experiment.

The outcomes are mutually exclusive.

The random variable is the result of counts.

Each trial is independent of any other trial.

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LO 6-5 Describe and compute probabilities for a binomial distribution.

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Binomial Probability Experiment

1. An outcome on each trial of an experiment is classified into one of two mutually exclusive categories—a success or a failure.

  • The random variable counts the number of successes in a fixed number of trials.
  • The probability of success and failure stay the same for each trial.
  • The trials are independent, meaning that the outcome of one trial does not affect the outcome of any other trial.

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LO 6-5

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Binomial Probability Formula

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LO 6-5

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Binomial Probability – Example

There are five flights daily from Pittsburgh via US Airways into the Bradford Regional Airport in PA. Suppose the probability that any flight arrives late is .20.

What is the probability that none of the flights are late today?

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LO 6-5

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Binomial Probability – Excel

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LO 6-5

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Binomial Distribution – Mean and Variance

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LO 6-5

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For the example regarding the number of late flights, recall that  = .20 and n = 5.

What is the average number of late flights?

What is the variance of the number of late flights?

Binomial Distribution – Mean and Variance: Example

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LO 6-5

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Binomial Distribution – Mean and Variance: Another Solution

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LO 6-5

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Binomial Distribution – Table

In a region of a country, five percent of all cell phone calls are dropped. What is the probability that out of six randomly selected calls, none were dropped?

Given Data:

n = 6 (sample size)

π = 0.05 (probability of success – dropped call)

x = 0 (number of dropped calls)

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LO 6-5

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Binomial Distribution – Table

Given Data:

n = 6, π = 0.05, x = 0

Find P(x = 0) =?

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LO 6-5

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Binomial Distribution – Table

Given Data:

n = 6, π = 0.05, x = 0

What is the probability that out of six randomly selected calls exactly one, exactly two, exactly three, exactly four, exactly five, or exactly six are dropped calls?

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LO 6-5

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Binomial Distribution – MegaStat

In a region of a country, five percent of all cell phone calls are dropped. What is the probability that out of six randomly selected calls, …

None will be dropped?

Exactly one?

Exactly two?

Exactly three?

Exactly four?

Exactly five?

Exactly six out of six?

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LO 6-5

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Binomial – Shapes for Varying  (n constant)

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LO 6-5

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Binomial – Shapes for Varying n ( constant)

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LO 6-5

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Binomial Probability Distributions – Example

A study by the Illinois Department of Transportation concluded that 76.2 percent of front seat occupants used seat belts. A sample of 12 vehicles is selected.

What is the probability the front seat occupants in exactly 7 of the 12 vehicles are wearing seat belts?

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LO 6-5

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Binomial Probability Distributions – Example

Given Data:

n = 12 vehicles (sample size)

π = 0.762 (probability of success – wearing seatbelt)

x = 7 (front seat occupants wearing seatbelts)

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LO 6-5

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Binomial Probability Distributions – Example

Given Data:

n = 12 vehicles

π = 0.762 (proportion wearing seatbelt)

What is the probability the front seat occupants in at least 7 of the 12 vehicles are wearing seat belts?

P(x ≥ 7) = ?

P(x =7,8,9,10,11,12) = ?

P(x ≥ 7) = 0.9562

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LO 6-5

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Cumulative Binomial Probability Distributions – Excel

=binomdist(6,12,0.762,0)

=1- binomdist(6,12,0.762,1)

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LO 6-5

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Poisson Probability Distribution

The Poisson probability distribution describes the number of times some event occurs during a specified interval. The interval may be time, distance, area, or volume.

Assumptions of the Poisson Distribution

  • The probability is proportional to the length of the interval.
  • The intervals are independent.

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LO 6-6 Describe and compute probabilities for a Poisson distribution.

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Poisson Probability Distribution

The Poisson probability distribution is characterized by the number of times an event happens during some interval or continuum.

Examples include:

• The number of misspelled words per page in a newspaper.

• The number of calls per hour received by Dyson Vacuum Cleaner Company.

• The number of vehicles sold per day at Hyatt Buick GMC in Durham, North Carolina.

• The number of goals scored in a college soccer game.

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LO 6-6

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Poisson Probability Distribution

The Poisson distribution can be described mathematically using the formula:

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LO 6-6

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Poisson Probability Distribution

  • The mean number of successes μ can be determined in Poisson situations by n, where n is the number of trials and  the probability of a success.

  • The variance of the Poisson distribution is also equal to n.

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LO 6-6

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Poisson Probability Distribution – Example

Assume baggage is rarely lost by Northeast Airlines. Suppose a random sample of 1,000 flights shows a total of 300 bags were lost. Thus, the arithmetic mean number of lost bags per flight is 0.3 (300/1,000). If the number of lost bags per flight follows a Poisson distribution with u = 0.3, find the probability of not losing any bags.

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LO 6-6

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Poisson Probability Distribution – Table

Recall from the previous illustration that the number of lost bags follows a Poisson distribution with a mean of 0.3. Use Appendix B.5 to find the probability that no bags will be lost on a particular flight. What is the probability no bag will be lost on a particular flight?

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LO 6-6

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More About the Poisson Probability Distribution

  • The Poisson probability distribution is always positively skewed and the random variable has no specific upper limit.

  • The Poisson distribution for the lost bags illustration, where µ=0.3, is highly skewed.
  • As µ becomes larger, the Poisson distribution becomes more symmetrical.

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LO 6-6

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