Transcript Slides for Chapter 6

6- 1
Chapter
Six
McGraw-Hill/Irwin
© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.
Chapter Six
6- 2
Discrete Probability Distributions
GOALS
When you have completed this chapter, you will be able to:
ONE
Define the terms random variable and probability
distribution.
TWO
Distinguish between a discrete and continuous probability
distributions.
THREE
Calculate the mean, variance, and standard deviation of a
discrete probability distribution.
Chapter Six
6- 3
continued
Discrete Probability Distributions
GOALS
When you have completed this chapter, you will be able to:
FOUR
Describe the characteristics and compute probabilities
using the binomial probability distribution.
FIVE
Describe the characteristics and compute probabilities
using the hypergeometric distribution.
SIX
Describe the characteristics and compute the
probabilities using the Poisson distribution.
6- 4
Probability
Distribution
Types of Probability
Distributions
A listing of all
possible outcomes
Discrete probability Distribution
of an experiment
Can assume only certain
a n d
t h e
outcomes
corresponding
p r o b a b i l i t y . Continuous Probability Distribution
Random variable Can assume an infinite number of
values within a given range
A numerical value
determined by the
outcome of an
experiment.
Types of Probability Distributions
6- 5
Continuous Probability Distribution
Movie
6- 6
Discrete Probability Distribution
The sum of the
probabilities of
the various
outcomes is 1.00.
The outcomes
are mutually
exclusive.
The probability
of a particular
outcome is
between 0 and
1.00.
The number of
students in a class
The number of
cars entering a
carwash in a hour
The number of
children in a family
Features of a Discrete Distribution
6- 7
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.
From the definition
of a random variable,
x as defined in this
experiment, is a
random variable.
The possible outcomes for
such an experiment
TTT, TTH, THT, THH,
HTT, HTH, HHT, HHH
Thus the possible
values of x (number
of heads) are 0,1,2,3.
Example 1
The
outcome of
two heads
occurred
three times.
The
outcome of
three heads
occurred
once.
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The outcome of zero
heads occurred once.
The
outcome
of one
head
occurred
three
times.
EXAMPLE 1
continued
6- 9
The long-run average value
of the random variable
Mean
The central location of the data
  [ xP( x)]
A weighted average
Also referred to as its
expected value, E(X), in a
probability distribution
where
 represents the mean
P(x) is the probability of
the various outcomes x.
The Mean of a Discrete
Probability Distribution
6- 10
Variance
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.
  [(x   ) P( x)]
2
2
The Variance of a Discrete
Probability Distribution
# houses
Painted
# of
weeks
Percent
of weeks
10
5
20 (5/20)
11
6
30 (6/20)
6- 11
Physics
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.
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# houses Probability
painted (x)
P(x)
10
.25
x*P(x)
2.5
11
.30
3.3
12
.35
4.2
13
.10
1.3

11.3
Mean number of
houses painted
per week
  [ xP( x)]
EXAMPLE 2
6- 13
Variance in the number of
2
2



[(
x


)
P( x)]
houses painted per week
# houses Probability
painted (x)
P(x)
(x-)
(x-)2
(x-)2
P(x)
10
11
.25
.30
10-11.3
11-11.3
1.69
.09
.423
.027
12
13
.35
.10
12-11.3
13-11.3
.49
2.89
2 
.171
.289
.910
6- 14
Binomial 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.
The trials are independent.
Binomial Probability
Distribution
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Binomial Probability Distribution
P( x)n Cx (1   )
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
Cx 
n
n!
x!(n-x)!
Binomial Probability
Distribution
6- 16
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?
P( x  3)14 C3 (.20 ) 3 (.80 )11  ...14 C14 (.20 )14 (.80 ) 0
 .250  .172  ...  .000  .551
At least three are
unemployed?
P (3)14 C 3 (.20 ) 3 (1  .20 )11
 (364 )(. 0080 )(. 0859 )
 .2501
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The probability at least one is unemployed?
P( x  1)  1  P(0)
 114 C0 (.20 ) (1  .20 )
0
14
 1  .044  .956
Example 3
6- 18
Mean of the Binomial Distribution
  n
Variance of the Binomial Distribution
  n (1   )
2
Mean & Variance of the Binomial
Distribution
6- 19
Example 3 Revisited
Recall that =.2 and n=14
= n = 14(.2) = 2.8
2 = n (1-  ) = (14)(.2)(.8) =2.24
Mean and Variance Example
6- 20
Finite Population
A population consisting
of a fixed number of
known individuals,
objects, or measurements
The number
of students in
this class
The number of cars
in the parking lot
The number of homes
built in Blackmoor
Finite Population
6- 21
Hypergeometric Distribution
Only 2 possible outcomes
Results from a count of
the number of successes in
a fixed number of trials
Trials are not independent
Sampling from a finite population
without replacement, the probability of
a success is not the same on each trial.
Hypergeometric Distribution
6- 22
Hypergeometric Distribution
( S Cx )( N  S Cn  x )
P( x ) 
N Cn
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
Hypergeometric Distribution
6- 23
Use 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
Hypergeometric Distribution
6- 24
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.
What is the probability
that three of five
violations randomly
selected to be investigated
are actually violations?
EXAMPLE 5
6- 25
( 4 C 3 )( 10 4 C 5 2
P (3) 
10 C 5
( 4 C 3 )( 6 C 2 ) 4(15 )


 .238
252
10 C 5
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.
Poisson probability
distribution
6- 26
Poisson Probability Distribution
 e
x
P( x ) 
u
x!
 = np
where
n is the number of trials
p the probability of a
success
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
Variance
Also equal to np
Poisson probability
distribution
6- 27
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?
 e
4 e
P( x) 

 .1465
x!
2!
x u
2
4
EXAMPLE 6