Discrete Probability Distributions

Chapter 6

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Learning Objectives
LO6-1 Identify the characteristics of a probability
distribution
LO6-2 Distinguish between discrete and continuous
random variables
LO6-3 Compute the mean, variance, and standard
deviation of a discrete probability distribution
LO6-4 Explain the assumptions of the binomial
distribution and apply it to calculate
probabilities
LO6-5 Explain the assumptions of the hypergeometric
distribution and apply it to calculate probabilities
LO6-6 Explain the assumptions of the Poisson distribution
and apply it to calculate probabilities

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What is a Probability Distribution?

PROBABILITY DISTRIBUTION A listing of all the outcomes of an

experiment and the probability associated with each outcome.

CHARACTERISTICS OF A PROBABILITY DISTRIBUTION

1. The probability of a particular outcome is between 0 and 1
inclusive.
2. The outcomes are mutually exclusive.
3. The list of outcomes is exhaustive. So the sum of the probabilities of
the outcomes is equal to 1.

 Example: A drug manufacturer may claim a treatment will

cause weight loss for 80% of the population. This claim could
be tested by a consumer protection agency using a sample and
statistical inference.

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Probability Distribution Example
 Suppose we are interested in the number of heads
showing face up with 3 tosses of a coin
 The possible outcomes are 0 heads, 1 head, 2 heads, and 3
heads

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Probability Distribution Table
 Probability distribution table and chart for the events of
zero, one, two, and three heads

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Random Variables
 In any experiment of chance, the outcomes occur
randomly, and so are called random variables

RANDOM VARIABLE A quantity resulting from an experiment that, by

chance, can assume different values.

 Examples
 The number of employees absent from the day shift on
Monday, the number might be 0, 1, 2, 3, …The number
absent is the random variable
 The grade level (Freshman, Sophomore, Junior, or Senior)
of the members of the St. James High School Varsity girls’
basketball team. Grade level is the random variable
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Two Types of Random Variables
 One type of random variable is the discrete random
variable
 Discrete variables are usually the result of counting
DISCRETE RANDOM VARIABLE A random variable that can assume
only certain clearly separated values.

 Examples
 Tossing a coin three times and counting the number of
heads
 A bank counting the number of credit cards carried by a
group of customers

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Discrete Random Variable
 For example, the Bank of the Carolinas counts the
number of credit cards carried by a group of customers
 The number of cards carried is the discrete random
variable

Number of Credit Cards Relative Frequency

0 .03
1 .10
2 .18
3 .21
4 or more .48
Total 1.00

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Continuous Random Variables
 Continuous random variables can assume an infinite
number of values within a given range
 Continuous variables are usually the result of measuring
 Examples
 The time between flights between Atlanta and LA are
4.67 hours, 5.13 hours, and so on
 The annual snowfall in Minneapolis, MN measured in
inches

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Mean and Variance of a Probability
Distribution
 The mean is a typical value used to represent the central
location of the data
 The mean is also referred to as the expected value

 The amount of spread (or variation) in the data is

described by the variance

 The standard deviation of the probability distribution is

the positive square root of the variance

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Probability Distribution Mean Example
John Ragsdale sells new
cars for Pelican Ford. John
usually sells the most cars
on Saturday. He has 1. What type of distribution is this?
developed a probability 2. How many cars does John expect
distribution for the to sell on a typical Saturday?
number of cars he expects 3. What is the variance?
to sell on Saturday.

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Probability Distribution Variance Example
 The computational steps for variance
 Subtract the mean from each value of x and square
 Multiply each squared difference by its probability
 Sum the resulting products to arrive at the variance

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Binomial Distribution
 There are four requirements of a binomial probability
distribution
1. There are only two possible outcomes and the outcomes are
mutually exclusive, as either a success or a failure
2. The number of trials is fixed and known
3. The probability of a success is the same for each trial
4. Each trial is independent of any other trial

 Example
 A young family has two children, both boys. The
probability of the third birth being a boy is still .50. The
gender of the third child is independent of the gender of
the other two.

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Binomial Probability Experiment
 Use the number of trials, n, and the probability of a
success, π, to compute binomial probability

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.
2. The random variable is the number of successes in a fixed number
of trials.
3. The probability of success is the same for each trial.
4. The trials are independent, meaning that the outcome of one trial
does not affect the outcome of any other trial.

 Note: Do not confuse the symbol π, with the

mathematical constant 3.1416

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How is a Binomial Probability Computed?

There are five flights daily from Pittsburgh via US Airways into the Bradford
Regional Airport in Bradford, Pennsylvania. Suppose the probability that any
flight arrives late is .20.
What is the
probability that none P(x) = nCr(π)r 1 − π n − r
of the flights are late P(0) = 5C0(.20)0 1 − .20 5 − 0
today? = (1)(1)(.3277) = .3277

What is the
probability that P(x) = nCr(π)r 1 − π n − r
exactly one of the P(1) = 5C1(.20)1 1 − .20 5 − 1
flights is late today? = (1)(1)(.4096) = .4096

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Binomial Probability Distribution

There are five flights daily from Pittsburgh via US Airways into the Bradford
Regional Airport in Bradford, Pennsylvania. Suppose the probability that any
flight arrives late is .20. What is the probability that none of the flights are late
today? What is the probability that exactly 1 of the flights is late today?

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Shortcut Formulas

 Using the preceding example of flights into Bradford

Airport; n=5 and π = .20 and the shortcut formulas
μ=nπ
μ = 5 .20 = 1.00
σ2 = nπ 1 − π
σ2 = 5 .20 1 − .20 = .80

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Binomial Probability Tables
 Tables are already constructed for use as well
In the Southwest, 5% of all cell phone calls are dropped. What is the
probability that out of six randomly selected calls, none was dropped?
Exactly one? Exactly two? Exactly three? Exactly four? Exactly five?
Exactly six out of six? See the table below for the answers.

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Cumulative Binomial Probability
Distributions
A study by the Illinois Department of Transportation concluded that 76.2% of
front seat occupants wore seat belts. That is, both occupants of the front seat
were using their seat belts. Suppose we decide to compare that information with
current usage. We select a sample of 12 vehicles.

1. What is the probability that the front seat occupants in exactly 7 of the 12
vehicles are wearing seat belts?
P(x) = nCr(π)r 1 − π n−r

P(x=7) = 12C7(.762)7 1 − .762 12 − 7

= 792(.149171)(.000764) = .0902
2. What is the probability that at least 7 of the 12 front seat occupants are
wearing seat belts?
P(x≥7) = P(x=7) + P(x=8) + P(x=9) + P(x=10) + P(x=11) + P(x=12)
=.0902 + .1805 + .2569 + .2467 + .1436 + .0383
=.9562

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Hypergeometric Distribution
 When sampling from relatively small populations without
replacement, use the hypergeometric distribution

HYPERGEOMETRIC 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.
2. The random variable is the number of successes in a fixed number
of trials.
3. The trials are not independent.
4. We assume that we sample from a finite population without
replacement and n/N > 0.05. So, the probability of a success changes
for each trial.

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Hypergeometric Formula

PlayTime Toys Inc. employs 50

(40𝐶4)(50−40𝐶5−4)
people in the Assembly Dept. Forty 𝑃 4 =
of the employees belong to a union 50𝐶5
and 10 do not. Five employees are (91,390)(10)
selected at random to form a = = .431
2,118,760
committee. What is the probability
that four of the five belong to a
union.

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Hypergeometric Probabilities

Union Members Probability

0 .000
1 .004
2 .044
3 .210
4 .431
5 .311
1.000

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Poisson Probability Distribution
 This describes the number of times some event occurs
during a specified interval
 The interval can be time, distance, area, or volume
 Two assumptions
 The probability is proportional to the length of the
interval
 The intervals are independent
 The Poisson has many applications like describing
 The distribution of errors in data entry
 The number of accidents on I-75 during a three-month
period
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Poisson Distribution
POISSON PROBABILITY EXPERIMENT
1. The random variable is the number of times some event occurs
during a defined interval.
2. The probability of the event is proportional to the size of the
interval.
3. The intervals do not overlap and are independent.

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Poisson Distribution Example

Budget Airlines is a seasonal airline that operates flights from Myrtle Beach,
South Carolina, to various cities in the northeast. Recently Budget has been
concerned about the number of lost bags. Ann Poston from the Analytics
Department was asked to study the issue. She randomly selected a sample of
500 flights and found that a total of twenty bags were lost on the sampled flights.
The mean number of bags lost, μ, is found by 20/500 = .04

The probability that no bags are lost is found using formula 6-7.
μxe−μ .040𝑒 −0.04
P 0 = = = .9608
x! 0!

Then calculate the probability that one or more bags is lost.

μxe−μ .040𝑒 −0.04
P(x≥1) =1-P 0 = 1 − =1- = 1- .9608 = .0392
x! 0!

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Poisson Probability Distribution Tables
NewYork-LA Trucking company finds the mean number of breakdowns on
the New York to Los Angeles route is 0.30. From the table, we can locate
the probability of no breakdowns on a particular run. Find the column 0.3,
then read down that column to the row labeled 0; the value is .7408. The
probability of 1 breakdown is .2222

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