mth120_chapter6

Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin

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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.

FOURDescribe the characteristics and compute probabilities using the binomial probability distribution.


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Chapter Six continuedA Survey of Probability A Survey of Probability ConceptsConceptsGOALS

When you have completed this chapter, you will be able to:

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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Random VariablesRandom Variables

A random variable is a numerical value determined by the outcome of an experiment.

A probability distribution is the listing of all possible outcomes of an experiment and the corresponding probability.


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Types of Probability DistributionsTypes of Probability Distributions

A discrete probability distribution can assume only certain outcomes.

A continuous probability distribution can assume an infinite number of values within a given range.


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Examples of a discrete distribution are:

The number of students in a class.The number of children in a family.The number of cars entering a carwash in a hour.Number of home mortgages approved by Coastal

Federal Bank last week.


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Examples of a continuous distribution include:

The distance students travel to class.The time it takes an executive to drive to work.The length of an afternoon nap.The length of time of a particular phone call.


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Features of a Discrete DistributionFeatures of a Discrete Distribution

The main features of a discrete probability distribution are:

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.


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Example 1Example 1

EXAMPLE 1: 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 EXAMPLE 1 continuedcontinued

The possible outcomes for such an experiment will be:

TTT, TTH, THT, THH,

HTT, HTH, HHT, HHH.

Thus the possible values of x (number of heads) are 0,1,2,3.


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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.From the definition of a random variable, x as

defined in this experiment, is a random variable.


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The Mean of a Discrete Probability The Mean of a Discrete Probability DistributionDistribution

The mean: reports the central location of the data.is the long-run average value of the random variable.is also referred to as its expected value, E(X), in a

probability distribution.is a weighted average.


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The Mean of a Discrete Probability The Mean of a Discrete Probability DistributionDistribution

The mean is computed by the formula:

where represents the mean and P(x) is the probability of the various outcomes x.

)]([ xxP


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The Variance of a Discrete The Variance of a Discrete Probability DistributionProbability Distribution

The variance measures the amount of spread (variation) of a distribution.

The variance of a discrete distribution is denoted by the Greek letter 2 (sigma squared).

The standard deviation is the square root of 2.


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The Variance of a Discrete The Variance of a Discrete Probability DistributionProbability Distribution

The variance of a discrete probability distribution is computed from the formula:

)]()[( 22 xPx


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EXAMPLE 2EXAMPLE 2

Dan Desch, owner of College Painters, studied his records for the past 20 weeks and reports the following number of houses painted per week:

# o f H o u s e sP a i n t e d

W e e k s

10 5

11 6

12 7

13 2


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Probability Distribution:

Number of houses painted, x

Probability, P(x)

10 .25

11 .30

12 .35

13 .10

Total 1.00


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Compute the mean number of houses painted per week:

3.11

)10)(.13()35)(.12()30)(.11()25)(.10(

)]([)(

xxPxE


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Compute the variance of the number of houses painted per week:

91.0

2890.01715.00270.04225.0

)10(.)3.1113(...)25(.)3.1110(

)]()[(22

22

xPx


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Binomial Probability DistributionBinomial Probability Distribution

The binomial distribution has the following characteristics:

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.The trials are independent.


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To construct a binomial distribution, let n be the number of trials x be the number of observed successes be the probability of success on each trial


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The formula for the binomial probability distribution is:

xnxxn CxP )1()(


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EXAMPLE 3EXAMPLE 3

The Alabama Department of Labor reports that 20% of the workforce in Mobile is unemployed. From a sample of 14 workers, calculate the following probabilities:Exactly three are unemployed.At least three are unemployed.At least none are unemployed.


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The probability of exactly 3:

The probability of at least 3 is:

2501.

)0859)(.0080)(.364(

)20.1()20(.)3( 113314

CP

551.000....172.250.

)80(.)20(....)80(.)20(.)3( 0141414

113314

CCxP


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Example 3 Example 3 continuedcontinued

The probability of at least one being unemployed.

956.044.1

)20.1()20(.1

)0(1)1(140

014

C

PxP


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Mean & Variance of the Binomial Mean & Variance of the Binomial DistributionDistribution

The mean is found by:

The variance is found by:

n

2 1 n ( )


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EXAMPLE 4EXAMPLE 4

From EXAMPLE 3, recall that =.2 and n=14.

Hence, the mean is:

= n = 14(.2) = 2.8.

The variance is:

2 = n (1- ) = (14)(.2)(.8) =2.24.


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Finite PopulationFinite Population

A finite population is a population consisting of a fixed number of known individuals, objects, or measurements. Examples include:

The number of students in this class.The number of cars in the parking lot.The number of homes built in Blackmoor.


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Hypergeometric DistributionHypergeometric Distribution

The hypergeometric distribution has the following characteristics:

There are only 2 possible outcomes.The probability of a success is not the same on each trial.It results from a count of the number of successes in a

fixed number of trials.


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Hypergeometric DistributionHypergeometric DistributionThe formula for finding a probability using the

hypergeometric distribution is:

where N is the size of the population, S is the number of successes in the population, x is the number of successes in a sample of n observations.

P xC C

CS x N S n x

N n

( )( )( )


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Hypergeometric DistributionHypergeometric Distribution

Use the hypergeometric distribution to find the probability of a specified number of successes or failures if: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).

the size of the sample n is greater than 5% of the size of the population N .


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EXAMPLE 5EXAMPLE 5

The National Air Safety Board has a list of 10 reported safety violations. Suppose only 4 of the reported violations are actual violations and the Safety Board will only be able to investigate five of the violations. What is the probability that three of five violations randomly selected to be investigated are actually violations?


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238.252

)15(4))((

)(()3(

510

2634

510

2541034

C

CC

C

CCP


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Poisson Probability DistributionPoisson Probability Distribution

The binomial distribution becomes more skewed to the right (positive) as the probability of success become smaller.

The limiting form of the binomial distribution where the probability of success is small and n is large is called the Poisson probability distribution.


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The Poisson distribution can be described mathematically using the formula:

where is the mean number of successes in a particular interval of time, e is the constant 2.71828, and x is the number of successes.

P xe

x

x u

( )!


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The mean number of successes can be determined in binomial 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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EXAMPLE 6EXAMPLE 6

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 4 arrivals in an hour?

1465.!2

4

!)(

42

e

x

exP

ux

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