Transcript Binomial vs. Geometric

Binomial vs. Geometric
Chapter 8
Binomial and Geometric
Distributions
Part 2
Binomial vs. Geometric
The Binomial Setting The Geometric Setting
1. Each observation falls into 1. Each observation falls into
one of two categories.
one of two categories.
2. The probability of success 2. The probability of success
is the same for each
is the same for each
observation.
observation.
3. The observations are all
3. The observations are all
independent.
independent.
4. There is a fixed number n
of observations.
4. The variable of interest is the
number of trials required to
obtain the 1st success.
Are Random Variables and Binomial
Distributions Linked?
X = number of people who purchase electric hot tub
X 0
1
2
3
P(X) .216 .432 .288 .064
GGG
(.6)(.6)(.6)
EGG
GEG
GGE
(.4)(.6)(.6)
(.6)(.4)(.6)
(.6)(.6)(.4)
EEG
GEE
EGE
(.4)(.4)(.6)
(.6)(.4)(.4)
(.4)(.6)(.4)
EEE
(.4)(.4)(.4)
Combinations
Formula:
nI
n!
F

G
J
Hk K k !an  k f!
Practice:
6
6!
6  5 4  3 2 1 6  5
 15


1.

2 1
4! 6  4 ! 4  3  2  1 2  1
4
F
I
G
J
HK a f
af
8I 8! 8  7  6  5!
F


2. G
 56
J
H5K 5!3! 5! 3  2 1
Developing the Formula
X
2
3
0
1
P( X ) .4219 .4219 .1406 .0156
Outcomes
Probability
Rewritten
3
0
3
c
c
c
1
.
25
.
75
(.75)(.75)(.75)
OOO
0
1
2
OOcOc
3
(.
25
)(.
75
)(.
75
)
3 .25 .75
c
c
O OO
1
c
c
OOO
OOOc
OOcO
OcOO
OOO
(.25)(.25)(.75)
(.25)(.25)(.25)
F
I
G
J
a
fa
f
HK
F
I
a
fa
f
G
J
HK
3I
F
.25fa
.75f
3a
G
J
H2K
3I
F
.25f a
.75f
1a
G
J
H3K
2
1
3
0
Developing the Formula
n = # of observations
p = probablity of success
k = given value of variable
nI
F
p a
P( X  k )  G
1 pf
J
Hk K
k
F
I
G
J
a
fa
f
HK
3I
F
.25fa
.75f
3a
G
J
H1K
3I
F
.25fa
.75f
3a
G
J
H2K
3I
F
.25f a
.75f
1a
G
J
H3K
Rewritten
0
3
1
.
25
.
75
0
3
n k
1
2
2
1
3
0
Working with probability
distributions
State the distribution to be used
Define the variable
State important numbers
Binomial: n & p
Geometric: p
Twenty-five percent of the customers entering a
grocery store between 5 p.m. and 7 p.m. use an
express checkout. Consider five randomly selected
customers, and let X denote the number among the
five who use the express checkout.
binomial
n=5
p = .25
X = # of people use express
What is the probability that two used express
checkout?
binomial
n=5
p = .25
X = # of people use express
5
2
3
P  X  2     .25  .75   .2637
 2
What is the probability that at least four used
express checkout?
binomial
n=5
p = .25
X = # of people use express
5
5
4
1
5
P  X  4     .25  .75     .25 
5
 4
 .0156
“Do you believe your children will have a higher
standard of living than you have?” This question was
asked to a national sample of American adults with
children in a Time/CNN poll (1/29,96). Assume that
the true percentage of all American adults who
believe their children will have a higher standard of
living is .60. Let X represent the number who believe
their children will have a higher standard of living
from a random sample of 8 American adults.
binomial
n=8
p = .60
X = # of people who believe…
Interpret P(X = 3) and find the numerical answer.
binomial
n=8
p = .60
X = # of people who believe
The probability that 3 of the people from the
random sample of 8 believe their children will
have a higher standard of living.
8
3
5
P  X  3    .6  .4 
 3
 .1239
Find the probability that none of the parents
believe their children will have a higher standard.
binomial
n=8
p = .60
X = # of people who believe
8
0
8
P  X  0     .6  .4 
0
 .00066