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Chapter
Six
McGraw-Hill/Irwin
© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.
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Chapter SixDiscrete Probability Discrete Probability DistributionsDistributionsGOALS
When you have completed this chapter, you will be able to:ONEDefine the terms random variable and probability distribution.
TWO Distinguish between a discrete and continuous probability distributions.
THREECalculate the mean, variance, and standard deviation of a discrete probability distribution.
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Chapter Six continued
Discrete Probability Discrete Probability DistributionsDistributionsGOALS
When you have completed this chapter, you will be able to:FOURDescribe the characteristics and compute probabilities using the binomial probability distribution.
FIVE Describe the characteristics and compute probabilities using the hypergeometric distribution.
SIXDescribe the characteristics and compute the probabilities using the Poisson distribution.
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Types of Probability Distributions
Discrete probability DistributionDiscrete probability Distribution Can assume only certain
outcomes
Continuous Probability DistributionContinuous Probability Distribution Can assume an infinite number of
values within a given range
Types of Probability Types of Probability DistributionsDistributions
Probability Probability DistributionDistribution
A listing of all possible outcomes of an experiment and the corresponding probability.
Random variableRandom variable A numerical value determined by the
outcome of an experiment.
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Movie
Continuous Probability Distribution Continuous Probability Distribution
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Features of a Discrete Distribution
Discrete Probability Distribution Discrete Probability Distribution
The sum of the probabilities of
the various outcomes is 1.00.
The probability of a particular
outcome is between 0 and
1.00.
The outcomes are mutually
exclusive.
The number of students in a class
The number of children in a family
The number of cars entering a
carwash in a hour
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Thus the possible values of x (number of heads) are 0,1,2,3.
From the definition of a random variable, x as defined in this experiment, is a random variable.
TTT, TTH, THT, THH,HTT, HTH, HHT, HHH
Example 1
The possible outcomes for such an experiment
Consider a random experiment in which a coin is tossed three times. Let x be the number of heads. Let H represent the outcome of a head and T the outcome of a tail.
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EXAMPLE 1 continued
The outcome of zero heads occurred once.
The outcome
of one head
occurred three times.
The outcome of two heads occurred
three times.
The outcome of three heads occurred once.
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The Mean of a Discrete Probability Distribution
)]([ xxP
Mean Mean
The central location of the data
The long-run average value of the random variable
Also referred to as its expected value, E(X), in a probability distribution
A weighted average
where represents the mean P(x) is the probability of
the various outcomes x.
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The Variance of a Discrete Probability Distribution
VarianceVariance
Measures the amount of spread
(variation) of a distribution
Denoted by the Greek letter s2
(sigma squared)
Standard deviation is the square root of s2.
)]()[( 22 xPx
6- 11# houses
Painted
# of weeks
Percent of weeks
10 5 20 (5/20)
11 6 30 (6/20)
12 7 35 (7/20)
13 2 10 (2/20)
Total percent 100 (20/20)
Dan Desch, owner of College Painters, studied his records for the past 20 weeks and reports the following number of houses painted per week.
P hysics
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EXAMPLE 2
)]([ xxP
# houses painted (x)
Probability
P(x) x*P(x)
10 .25 2.5
11 .30 3.3
12 .35 4.2
13 .10 1.3
11.3
Mean number of houses painted
per week
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# houses painted (x)
Probability
P(x) (x- (x-
(x-P(x)
10 .25 10-11.3 1.69 .423
11 .30 11-11.3 .09 .027
12 .35 12-11.3 .49 .171
13 .10 13-11.3 2.89 .289
.910
)]()[( 22 xPx Variance in the number of houses painted per week
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Binomial Probability Distribution
The trials are independent.
Binomial Probability DistributionBinomial Probability Distribution
An outcome of an experiment is
classified into one of two mutually exclusive
categories, such as a success or
failure.
The data collected are the results of
counts.
The probability of success stays the same for
each trial.
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Binomial Probability Distribution
xnC n!x!(n-x)!
n is the number of trials x is the number of observed successes
p is the probability of success on each trial
xnxxnCxP )1()(
Binomial Probability DistributionBinomial Probability Distribution
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551.000....172.250.
)80(.)20(....)80(.)20(.)3( 0141414
113314
CCxP
The Alabama Department of Labor reports that 20% of the workforce in Mobile
is unemployed and interviewed 14 workers.
What is the probability that exactly three are
unemployed?
At least three are unemployed?
2501.
)0859)(.0080)(.364(
)20.1()20(.)3( 113314
CP
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Example 3
956.044.1
)20.1()20(.1
)0(1)1(140
014
C
PxP
The probability at least one is unemployed?
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Mean & Variance of the Binomial Distribution
n
2 1 n ( )
Mean of the Binomial DistributionMean of the Binomial Distribution
Variance of the Binomial DistributionVariance of the Binomial Distribution
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Example 3 Revisited
Mean and Variance Example
Recall that =.2 and n=14
= n = 14(.2) = 2.8
2 = n (1- ) = (14)(.2)(.8) =2.24
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Finite Population
The number of homes built in Blackmoor
A population consisting of a fixed number of known individuals, objects, or measurements
Finite PopulationFinite Population
The number of students in this class
The number of cars in the parking lot
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Hypergeometric Distribution
Hypergeometric DistributionHypergeometric Distribution
Only 2 possible outcomes
Sampling from a finite population without replacement, the probability of a success is not the same on each trial.
Results from a count of the number of successes in
a fixed number of trials
Trials are not independent
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Hypergeometric Distribution
P xC C
CS x N S n x
N n
( )( )( )
where N is the size of the populationS is the number of successes in the populationx is the number of successes in a sample of n
observations
Hypergeometric DistributionHypergeometric Distribution
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Hypergeometric Distribution
The size of the sample n is greater than 5% of the size of the population N
The sample is selected from a finite population without replacement (recall that a criteria for the binomial distribution is that the probability of success remains the same from trial to trial).
Use to find the probability of a specified number of successes or failures if
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EXAMPLE 5
What is the probability that three of five violations randomly selected to be investigated are actually violations?
The Transportation Security Agency has a list of 10 reported safety violations. Suppose only 4 of the reported violations are actual violations and the Security Agency will only be able to investigate five of the violations.
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The limiting form of the binomial distribution where the probability of success is small and n is large is called the Poisson probability distribution.
The binomial distribution becomes more skewed to the right (positive) as the probability of success become smaller.
238.252
)15(4))((
)(()3(
510
2634
510
2541034
C
CC
C
CCP
Poisson probability distribution
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Poisson probability distribution
P xe
x
x u
( )!
where is the mean number of successes
in a particular interval of time e is the constant 2.71828 x is the number of successes
where
n is the number of trials
p the probability of a success
Variance Also equal to np
Poisson Probability DistributionPoisson Probability Distribution
= np
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EXAMPLE 6
1465.!2
4
!)(
42
e
x
exP
ux
The Sylvania Urgent Care facility specializes in caring for minor injuries, colds, and flu. For the evening hours of 6-10 PM the mean number of arrivals is 4.0 per hour. What is the probability of 2 arrivals in an hour?
