Binomial vs. Geometric
Chapter 8
Binomial and Geometric
Distributions
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.
For each of the following situations, determine if
the situation is binomial, geometric, or neither.
1.
You observe the gender of the next 20 children born at
a local hospital; X is the number of girls among them.
2.
Bobby draws cards from a deck until he gets an ace.
He then reshuffles, and starts over and repeats this
process until he “wins” 10 times. The count X is the
total number of cards he counted until he finished.
3.
Joe buys a “Texas 2-Step” lottery ticket every week
until he wins. He has a probability of .0023 of winning
in any given week.
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.
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 Binomial Formula
X
P(O) = .25
2
3
0
1
P(Oc) = .75
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
3
OOOc
2
1
(.25)(.25)(.75)
3 .25 .75
OOcO
2
c
O OO
3
3
0
(.25)(.25)(.25)
1 .25 .75
OOO
3
F
I
G
J
a
fa
f
HK
F
I
a
fa
f
G
J
HK
F
I
a
fa
f
G
J
HK
F
I
a
f
a
f
G
J
HK
Developing the Binomial 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
The Mean and Standard Deviation of
a Binomial Random Variable
If a count X has the binomial distribution with
number of observations n and probability of a
success p…
Mean:
  np
Standard Deviation:   np 1  p 
Probability Distribution Function (pdf)
Given a discrete random variable
X, the pdf assigns a probability to
each value of X.
Cumulative distribution Function (cdf)
Given a random variable X, cdf
calculates the sum of the
probabilities for 0, 1, 2, …, X.
It calculates the probability of
obtaining at most X successes in
“n” trials.
Calculator Functions:
pdf: means X = ?
2nd Distr. -> 0 (binompdf)
binompdf(n, p, X)
cdf: means X ≤ ?
2nd Distr. -> A (binomcdf)
binomcdf(n, p, X)
Ex 1. The count X of children with type
O blood among 5 children whose parents
carry genes for both O and A blood
types is B(5, .25).
Find P(X = 2).
Find P(X < 4)
Ex 2. Angel is a basketball player who makes 75% of her free throws
over the course of a season. At last night’s game, she missed 5 of 7
free throws. Assume that each free throw is independent of one
another…studies on long sequences of free throws have found no
evidence that they are dependent.
Define the variable.
Complete the table.
----------------------------------------------------------------------X
----------------------------------------------------------------------P(X)
----------------------------------------------------------------------F(X)
What is the probability that Angel
makes at most 6 free throws in a row?
What is the probability that Angel makes
more than 3 free throw shots?
Find the mean and Standard Deviation of
the following random variable.
According to a recent Census Bureau report,
12.7% of Americans live below the poverty
level. Suppose you plan to sample at random
100 Americans and count the number of
people who live below the poverty level.
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.
Developing the Geometric Formula
X = wanted
outcome O
X
1
2
3
4
Probability of a Success: P(O) = 16
Probability of a Failure: P(Oc) = 5 6
Probability
1
a fa f
P X  n  1 p
6
d6id6i
d56id56id16i
5
5
5
1
d6id6id6id6i
5
1
n 1
p
The Mean and Standard
Deviation of a Geometric Random
Variable
If X is a geometric random variable with
probability of success p on each trial,
the expected value of the random
variable (the expected number of trials
to get the first success) is
1 p
1


2
p
p
Calculator Functions:
pdf: means X = ?
2nd Vars (Distr.) -> D (geometpdf)
geometpdf(p, X)
cdf: means X ≤ ?
2nd Vars (Distr.) -> E (geometcdf)
geometcdf(p, X)
Ex 2. Creating a Geometric Probability Distribution
table for X = number rolls of a die until a 4 occurs:
The probability of rolling a 4 on die = 1
6
X
1
2
3
P(X)
1
6
5
36
25
216
4
5
…
…
A geometric probability does not have a fixed ending value
“n”, therefore it is considered an infinite sequence.
Meaning the “x” goes on infinitely.
Ex 2. What is the probability that it takes
more than 6 rolls to observe a 4?
Ex 3. What is the probability that it takes
at least 3 rolls to observe a 4?
Find the mean and Standard Deviation of
the following random variable.
A potential buyer will sample videotapes from a
large lot of new videotapes. If she finds at
least one defective one, she’ll reject the entire
lot. If ten percent of the lot is defective, what
is the probability that she’ll find a defective
tape by the 4th videotape?