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BUS210
Business Statistics
Chapter 6
Discrete
Probability Distributions
Learning Objectives
In this section, you will learn about:
 Probability distributions and their properties
 Important discrete probability distributions:




The Binomial Distribution
The Poisson Distribution
Calculating the expected value and variance
Using the Binomial and Poisson distributions
to solve business problems
BUS210: Business Statistics
Discrete Probability Distributions - 2
Definitions

Random variable:
representation of a possible numerical value
from an uncertain event.

Discrete random variables: “How many”
produce outcomes that come from counting
(e.g. number of courses you are taking).

Continuous random variables: “How much”
produce outcomes that come from a measurement
(e.g. your annual salary, or your weight).
BUS210: Business Statistics
Discrete Probability Distributions - 3
Random Variables
Random
Variables
Discrete
Random Variable
Ch. 6
BUS210: Business Statistics
Continuous
Random Variable
Ch. 7
Discrete Probability Distributions - 4
Discrete Random Variables

Can only assume a countable number of values
Examples:

Rolling five dice (as in Yahtzee)
If X is the number of times that
a 6 shows up, then X could be
equal to 0, 1, 2, 3, 4, or 5

Flipping a coin 3 times.
If X is the number of times that
heads comes up, then X could
be equal to 0, 1, 2, or 3
BUS210: Business Statistics
Discrete Probability Distributions - 5
Discrete Random Variables

A probability distribution is…
a mutually exclusive listing of all possible numerical
outcomes for a variable and the probability of an
occurrence associated with each outcome.

Example for a discrete random variable:
BUS210: Business Statistics
Number of Classes Taken
Probability
1
0.20
2
0.10
3
0.40
4
0.24
5 or more
0.06
Notice that the
total probability
adds up to 1.00
Discrete Probability Distributions - 6
Discrete Random Variable
Probability Distribution
4 possible outcomes
Experiment:
Toss 2 Coins.
Let X = # heads.
T
T
T
H
H
H
BUS210: Business Statistics
T
H
Probability
Probability Distribution
0.50
0.25
0
X
1
2
X
Freq Probability
0
1
1/4 = 0.25
1
2
2/4 = 0.50
2
1
1/4 = 0.25
Total = 1.00
Discrete Probability Distributions - 7
Discrete Random Variable
Expected Value
Expected Value is the mean
of a discrete random variable
E(X) = m =
N
å X P(X )
i
i
Also known as the
Weighted Average
i=1

Example: Experiment of 2 Tossed Coins
i
X
P(X)
XiP(Xi)
X1
0
1/4 = 0.25
(0)(0.25) = 0.00
X2
1
2/4 = 0.50
(1)(0.50) = 0.50
X3
2
1/4 = 0.25
(2)(0.25) = 0.50
E(X) = 1.00
BUS210: Business Statistics
Discrete Probability Distributions - 8
Discrete Random Variable
Measuring Dispersion

Variance of a discrete random variable
N
s 2 = å[X i - E(X)]2 P(X i )
i=1

Standard Deviation of a discrete random variable
s = s2 =
N
2
[X
E(X)]
P(X i )
å i
i=1
Look familiar? This is ( X - m )
BUS210: Business Statistics
2
Discrete Probability Distributions - 9
Discrete Random Variable
Measuring Dispersion (continued)

Example:
standard deviation of 2 tossed coins
s= s =
2
N
2
[X
E(X)]
P(X i ) = 0.50 = 0.707
å i
i=1
X
E(X)
X-E(X)
(X-E(X))2
P(X)
(X-E(X))2P(X)
X1
0
1
-1
1
0.25
0.25
X2
1
1
0
0
0.50
0.00
X3
2
1
+1
1
0.25
0.25
∑= 0.50
BUS210: Business Statistics
Discrete Probability Distributions - 10
Probability Distributions
Probability
Distributions
Discrete
Probability
Distributions
Continuous
Probability
Distributions
Binomial
Normal
Poisson
Exponential
Ch. 6
BUS210: Business Statistics
Ch. 7
Discrete Probability Distributions - 11
Probability Distributions
Binomial

A finite number of observations, n



20 tosses of a coin
12 auto batteries purchased from a wholesaler
Each observation is categorized as…

Success: “event of interest” has occurred, or



Failure: “event of interest” has not occurred


i.e. - heads (or tails) in each toss of a coin;
i.e. - defective (or not defective) battery
The complement of a success
These two categories, success & failure, are
mutually exclusive and collectively exhaustive
BUS210: Business Statistics
Note: A random
experiment with
only 2 outcomes is
known as a
Bernoulli
experiment.
Discrete Probability Distributions - 12
Probability Distributions
Binomial

The probability of…
 success (event occurring) is represented as π


failure (event not occurring) is 1 – π
Probability of success (π) is constant


(continued)
Probability of getting tails is the same for each toss of the coin
Observations are independent


Each event is unaffected by any other event
Two sampling methods deliver independence

Infinite population without replacement

Finite population with replacement
BUS210: Business Statistics
Discrete Probability Distributions - 13
Probability Distributions
Binomial

Applications:


(continued)
Situations where are there only two outcomes
Apple marks newly manufactured iPads as
either defective or acceptable
Visitors to Amazon’s website will either buy an
item or not buy an item.

Asking if voters will approve a referendum, a
pollster receives responses of “yes” or “no”

Federal Express marks a delivery as either
damaged or not damaged
BUS210: Business Statistics
Discrete Probability Distributions - 14
Probability Distributions
Binomial
(continued)
Counting Revisited:

You toss a coin three times.
In how many ways can you get two heads?

Possible ways: HHT, HTH, THH,
so there are three ways you can getting two heads.


This situation is fairly simple….but, what about 10
times?…a 100?...or even a thousand?
For more complicated situations we need a method
to be able to count the number of ways (combinations).
BUS210: Business Statistics
Discrete Probability Distributions - 15
Counting Techniques
Rule of Combinations

The number of combinations of
selecting x objects out of n objects is
n!
n Cx =
x!(n - x)!
where:
n! means “n factorial”
2! = (1)(2)
4! = (1)(2)(3)(4)
Note: 0! = 1 (by definition)
BUS210: Business Statistics
Discrete Probability Distributions - 16
Counting Techniques
Rule of Combinations

You visit Baskin & Robbins. How many different
possible 3 scoop combinations could you decide on for
your ice cream cone if you select from their 31 flavors?

The total choices is n = 31, and we select x = 3.
31!
31! 31·30 ·29 ·28!
=
=
31 C3 =
3!(31- 3)! 3!28!
3·2 ·1·28!
=
BUS210: Business Statistics
31·30 ·29
= 31·5·29 = 4495 combinations
3·2 ·1
Discrete Probability Distributions - 17
Probability Distributions
Binomial
n!
P(x) =
π x (1-π) n-x
n!(n-x)!
π = probability of “event of interest”
n = sample size (number of trials or observations)
x = number of “events of interest” in sample
for the number of heads in 3 coin flips, we would use…
π
= 0.5
n
=4
X
= can be 0, or 1, or 2, or 3
and
(1 – π) = 0.5
Each value of X will have
a different probability
æ nö
Note: other sources may show this equation as f (x) = ç ÷ p x (1- p)(n-x) or similar.
è xø
BUS210: Business Statistics
Discrete Probability Distributions - 18
Probability Distributions
Binomial Example
What is the probability of 2 successes in 7 observations if
the probability of the event of interest is 0.4?
x = 1, n = 7, and π = 0.4
n!
P(x = 2) =
p x (1- p )n-x
x!(n - x)!
7!
=
(0.4)2 (1- 0.4)7-2
2!(7 - 2)!
= (21)(0.4)2 (0.6)5
= 0.2613
BUS210: Business Statistics
Discrete Probability Distributions - 19
Probability Distributions
Binomial Example
The probability of a single customer purchasing
an extended warranty is 0.35.
What is the probability of having 3 of the next 10
customers purchase an extended warranty?
x=3 ; n = 10 ; p = 0.35
n!
so, P(X=3) =
p X (1- p )n-X
X!(n - X)!
10!
=
(.35)3 (1-.35)10-3
3!(10 - 3)!
= (120)(.042875)(0.04902)
= 0.2522
BUS210: Business Statistics
Discrete Probability Distributions - 20
Probability Distributions
Using the Binomial Table
n
x
…
π=.20
π=.25
π=.30
π=.35
π=.40
π=.45
π=.50
10
0
1
2
3
4
5
6
7
8
9
10
…
…
…
…
…
…
…
…
…
…
…
0.1074
0.2684
0.3020
0.2013
0.0881
0.0264
0.0055
0.0008
0.0001
0.0000
0.0000
0.0563
0.1877
0.2816
0.2503
0.1460
0.0584
0.0162
0.0031
0.0004
0.0000
0.0000
0.0282
0.1211
0.2335
0.2668
0.2001
0.1029
0.0368
0.0090
0.0014
0.0001
0.0000
0.0135
0.0725
0.1757
0.2522
0.2377
0.1536
0.0689
0.0212
0.0043
0.0005
0.0000
0.0060
0.0403
0.1209
0.2150
0.2508
0.2007
0.1115
0.0425
0.0106
0.0016
0.0001
0.0025
0.0207
0.0763
0.1665
0.2384
0.2340
0.1596
0.0746
0.0229
0.0042
0.0003
0.0010
0.0098
0.0439
0.1172
0.2051
0.2461
0.2051
0.1172
0.0439
0.0098
0.0010
10
9
8
7
6
5
4
3
2
1
0
…
π=.80
π=.75
π=.70
π=.65
π=.60
π=.55
π=.50
x
Examples:
BUS210: Business Statistics
n = 10, π = .35, x = 3
P(x = 3) = .2522
n = 10, π = .75, x = 2
P(x = 2) = .0004
Discrete Probability Distributions - 21
Probability Distributions
Shape of the Binomial
The shape of the binomial distribution…
depends on the values of π and n

n = 5 π = 0.5
n = 5 π = 0.1
P(X)
P(X)
.6
.4
.6
.4
.2
.2
X
0
0
1
BUS210: Business Statistics
2
3
4
5
0
X
0
1
2
3
4
5
Discrete Probability Distributions - 22
Binomial Distribution
Characteristics
m = E(x) = np

Mean

Variance and Standard Deviation
s = np (1- p )
2
s = np (1- p )
Where n = sample size
π = probability of the event of interest for any trial
(1 – π) = probability of no event of interest for any trial
BUS210: Business Statistics
Discrete Probability Distributions - 23
Binomial Distribution
Characteristics
EXAMPLES
m = np = (5)(.1) = 0.5
s = np (1- p )
= (5)(.1)(1- .1) = 0.6708
m = np = (5)(.5) = 2.5
s = np (1- p )
= (5)(.5)(1- .5) = 1.118
BUS210: Business Statistics
n = 5 π = 0.1
P(X)
.6
.4
.2
X
0
0
1
2
3
4
5
n = 5 π = 0.5
P(X)
.6
.4
.2
0
X
0
1
2
3
4
5
Discrete Probability Distributions - 24
Binomial Distribution
Characteristics
Compound Events
Individual probabilities can be added to obtain any desired combined
event probability.
Examples:
•The probability that less than 3 of the next 10 customers will purchase an
extended warranty is
P(x<3)
=
P(x=0)
+
P(x=1)
+
P(x=2)
= 0.0135 + 0.0725 + 0.1757 = 0.2617
•
The probability that 3 or fewer of the next 10 customers will purchase an
extended warranty is
P(x<3)
=
P(x=0)
+
P(x=1)
+
P(x=2)
+
P(x=3)
= 0.0135 + 0.0725 + 0.1757 + 0.2522 = 0.5139
BUS210: Business Statistics
Discrete Probability Distributions - 25
Binomial Distribution
Using Excel
Cell Formulas
nxπ
n x π x (1-π)
x
BUS210: Business Statistics
n
π
cum
Discrete Probability Distributions - 26
Probability Distributions
Probability
Distributions
Discrete
Probability
Distributions
Continuous
Probability
Distributions
Binomial
Normal
Poisson
Exponential
Ch. 6
BUS210: Business Statistics
Ch. 7
Discrete Probability Distributions - 27
Probability Distributions
The Poisson Distribution

Used when you are interested in…
the number of times an event occurs
within a continuous unit or interval.


Was actually
used in a study
of malaria
Such as time, distance, volume, or area.
Examples: The number of…
 potholes in mile of road (distance)
 computer crashes in a day (time)
 chocolate chips in a cookie (volume)
 mosquito bites on a person (area)
BUS210: Business Statistics
Discrete Probability Distributions - 28
Probability Distributions
The Poisson Distribution

Defining features:

Independent: The occurrence of any single event
does not impact the occurrence of any other event.

Constant: The average probability of an event
occurring in one area of opportunity remains the
same for all other similar areas. The average
number of events per unit is represented by 
(lambda)

The average probability of an event decreases or
increases proportionally to any decrease or increase
in the size of the area of opportunity.
BUS210: Business Statistics
Discrete Probability Distributions - 29
Probability Distributions
The Poisson Distribution
-λ λx
e
P(x) =
x!
where:
x = number of events in an area of opportunity
 = expected value (mean) of number of events
e = base of the natural logarithm system (2.71828...)
Note: other sources may show this equation as f (x) =
BUS210: Business Statistics
m x e- m
x!
or similar.
Discrete Probability Distributions - 30
Poisson Distribution
Characteristics


Mean
E(x) = m = l
Variance and Standard Deviation
Isn’t that
interesting?
s =l
2
s= l
where  = expected number of events
BUS210: Business Statistics
Discrete Probability Distributions - 31
Probability Distributions
Using the Poisson Table

X
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
0
1
2
3
4
5
6
7
0.9048
0.0905
0.0045
0.0002
0.0000
0.0000
0.0000
0.0000
0.8187
0.1637
0.0164
0.0011
0.0001
0.0000
0.0000
0.0000
0.7408
0.2222
0.0333
0.0033
0.0003
0.0000
0.0000
0.0000
0.6703
0.2681
0.0536
0.0072
0.0007
0.0001
0.0000
0.0000
0.6065
0.3033
0.0758
0.0126
0.0016
0.0002
0.0000
0.0000
0.5488
0.3293
0.0988
0.0198
0.0030
0.0004
0.0000
0.0000
0.4966
0.3476
0.1217
0.0284
0.0050
0.0007
0.0001
0.0000
0.4493
0.3595
0.1438
0.0383
0.0077
0.0012
0.0002
0.0000
0.4066
0.3659
0.1647
0.0494
0.0111
0.0020
0.0003
0.0000
Example: Find P(X = 2) if  = 0.50
e- l lX e-0.50 (0.50) 2
P(X = 2) =
=
= 0.0758
X!
2!
BUS210: Business Statistics
Discrete Probability Distributions - 32
Poisson Distribution
Using Excel
Cell Formulas
x
BUS210: Business Statistics
λ
cum
Discrete Probability Distributions - 33
Probability Distributions
Graphing the Poisson
0.70
0.60
0.50
X
P(X)
0
1
2
3
4
5
6
7
0.6065
0.3033
0.0758
0.0126
0.0016
0.0002
0.0000
0.0000
BUS210: Business Statistics
P(x)
 = 0.50
0.40
0.30
0.20
0.10
0.00
0
1
2
3
4
5
6
7
x
P(X=2) = 0.0758
Discrete Probability Distributions - 34
Probability Distributions
Shape of the Poisson
The shape of the Poisson Distribution
depends on the parameter  :

0.70
0.25
 = 0.50
0.60
 = 3.00
0.20
0.40
P(x)
P(x)
0.50
0.30
0.15
0.10
0.20
0.05
0.10
0.00
0.00
0
1
2
3
4
x
BUS210: Business Statistics
5
6
7
1
2
3
4
5
6
7
8
9
10
11
12
x
Discrete Probability Distributions - 35
Chapter Summary

Probability distributions

Discrete random variables

The Binomial distribution

The Poisson distribution
BUS210: Business Statistics
Discrete Probability Distributions - 36