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Sherlock Holmes once observed
that men are insoluble puzzles
except in the aggregate, where
they become mathematical
certainties.
“You can never foretell what any
one man will do,” observed
Holmes, “but you can say with
precision what an average number
will be up to.
Individuals vary, but
percentages remain constant.
So says the statistician.”
Basic Probability & Discrete
Probability Distributions
Why study Probability?
To infer something about the
population based on sample
observations
We use Probability
Analysis to measure
the chance that
something will occur.
What’s the chance
If I flip a coin it
will come up
heads?
50-50
If the probability of
flipping a coin is 50-50,
explain why when I
flipped a coin, six of
the tosses were heads
and four of the tosses
were tails?
Think of probability in the long
run:
A coin that is continually flipped, will
50% of the time be heads and
50% of the time be tails
in the long run.
Probability is a
proportion or fraction
whose values range between
0 and 1, inclusively.
The Impossible Event
Has no chance of
occurring and has
a probability of
zero.
The Certain Event
Is sure to occur
and has a
probability of
one.
Probability Vocabulary
1)
2)
3)
4)
5)
6)
7)
8)
Experiment
Events
Sample Space
Mutually Exclusive
Collectively Exhaustive
Independent Events
Compliment
Joint Event
Experiment
An activity for
which the
outcome is
uncertain.
Examples of an Experiment:
•
•
•
•
•
Toss a coin
Select a part for inspection
Conduct a sales call
Roll a die
Play a football game
Events
Each possible
outcome of
the
experiment.
Examples of an Event:
• Toss a coin
• Select a part for
inspection
• Conduct a sales
call
• Roll a die
• Play a football
game
• Heads or tails
• Defective or nondefective
• Purchase or no
purchase
• 1,2,3,4,5,or 6
• Win, lose, or tie
Sample Space
The set of ALL
possible outcomes of
an experiment.
Examples of Sample Spaces:
• Toss a coin
• Select a part for
inspection
• Conduct a sales
call
• Roll a die
• Play a football
game
• Heads, tails
• Defective,
nondefective
• Purchase, no
purchase
• 1,2,3,4,5,6
• Win, lose, tie
Mutually Exclusive Events
cannot both occur
simultaneously.
Collectively Exhaustive
A set of events is
collectively
exhaustive if one
of the events
must occur.
Independent Events
If the probability of one event
occurring is unaffected by
the occurrence or
nonoccurrence of the other
event.
Complement
The complement of Event A includes all events
that are not part of Event A.
The complement of Event A
is denoted by Ā or A’.
Example: The compliment of being male
is being female.
Joint Event
Has two or more characteristics.
Gender
Male (M)
Female (F)
Total
<30
(U)
60
40
100
Age (Years)
30-45
(B)
20
30
50
>45
(O)
40
10
50
Total
120
80
200
Probability Vocabulary
1)
2)
3)
4)
5)
Experiment
Events
Sample Space
Mutually Exclusive
Collectively
Exhaustive
6) Independent Events
7) Compliment
8) Joint Event
Quiz
What’s the difference between
Mutually Exclusive and
Collectively Exhaustive?
When you estimate a probability
You are estimating
the probability of
an EVENT
occurring.
When rolling two die, the probability
of rolling an 11 (Event A) is the
probability that Event A occurs.
It is written P(A)
P(A) = probability that event A occurs
With a sample space of the
toss of a fair die being
S = {1, 2, 3, 4, 5, 6}
Find the probability of the following
events:
1) An even number
2) A number less than or
equal to 4
3) A number greater than
or equal to 5.
Answers
1)P(even number) = P(2) + P(4) + P(6)=
1/6 + 1/6 + 1/6 = 3/6 =1/2
2)P(number ≤ 4) = P(1) + P(2) + P(3) + P(4)=
1/6 + 1/6 + 1/6 + 1/6 = 4/6 = 2/3
3)P(number ≥ 5) = P(5) + P(6) =
1/6 + 1/6 = 2/6 = 1/3
Approaches to Assigning
Probabilities
• The Relative Frequency
• The Classical Approach
• The Subjective Approach
Classical Approach to Assigning
Probability
Probability based on prior knowledge
of the process involved with each
outcome equally likely to occur in
the long-run if the selection process
is continually repeated.
Relative Frequency (Empirical)
Approach to Assigning Probability
Probability of an event occurring based on
observed data.
By observing an experiment n times, if Event
A occurs m times of the n times, the
probability that A will occur in the future is
P(A) = m /n
Example of Relative Frequency
Approach
1000 students take a probability
exam.
200 students score an A.
P(A) = 200/1000 = .2 or 20%
The Relative Frequency Approach assigned
probabilities to the following simple events
What is the probability a student will
pass the course with a C or
better?
P(A) = .2
P(B) = .3
P(C) = .25
P(D) = .15
P(F) = .10
Subjective Approach to Assigning
Probability
Probability based on
individual’s past
experience, personal
opinion, analysis of
situation. Useful if
probability cannot be
determined empirically.
We leave Base Camp; the Ascent
for the Summit Begins!
From a survey of 200 purchasers of a laptop
computer, a gender-age profile is
summarized below:
Male
Female
Total
CLASS FREQUENCY
120
80
200
Under 30
30 -45
Over 45
Total
CLASS FREQUENCY
100
50
50
200
These two categories (gender and age) can be summarized
together in a contingency or cross-tab table which allows
the viewer to see how these two categories interact
Male
Female
Total
CLASS FREQUENCY
120
80
200
Gender
Male (M)
Female (F)
Total
<30
(U)
60
40
100
Under 30
30 -45
Over 45
Total
Age (Years)
30-45
(B)
20
30
50
CLASS FREQUENCY
100
50
50
200
>45
(O)
40
10
50
Total
120
80
200
Marginal Probability
The probability that any one single
event will occur.
Example: P(M) = 120/200 = .6
Gender
Male (M)
Female (F)
Total
<30
(U)
60
40
100
Age (Years)
30-45
(B)
20
30
50
>45
(O)
40
10
50
Total
120
80
200
Gender
Male (M)
Female (F)
Total
<30
(U)
60
40
100
Age (Years)
30-45
(B)
20
30
50
>45
(O)
40
10
50
What’s the probability of being under 30?
What’s the probability of being female?
What’s the probability of being either
under 30 or over 45?
Total
120
80
200
Gender
Male (M)
Female (F)
Total
<30
(U)
60
40
100
Age (Years)
30-45
(B)
20
30
50
>45
(O)
40
10
50
Total
120
80
200
What is the complement of
being male? P(MC) or P(M’)
Joint Probability
The probability that both Events A and B will
occur.
This is written as P(A and B)
Gender
Male (M)
Female (F)
Total
<30
(U)
60
40
100
Age (Years)
30-45
(B)
20
30
50
>45
(O)
40
10
50
Total
120
80
200
Gender
Male (M)
Female (F)
Total
<30
(U)
60
40
100
Age (Years)
30-45
(B)
20
30
50
>45
(O)
40
10
50
Total
120
80
200
What is the probability of selecting a
purchaser who is female and under
age 30?
P(F and U) = 40/200 = .2 or 20%
Probability of A or B
The probability that either of two events
will occur.
This is written as P(A
or B).
Use the General Addition Rule which
eliminates double-counting.
General Addition Rule
P(A or B) = P(A) + P(B) – P(A and B)
Gender
Male (M)
Female (F)
Total
<30
(U)
60
40
100
Age (Years)
30-45
(B)
20
30
50
>45
(O)
40
10
50
Total
120
80
200
What is the probability of selecting a purchaser
who is male OR under 30 years of age?
P(M or U) = P(M) + P(U) – P(M and U)
=(120 + 100 – 60) / 200
= 160 / 200
= .8 or 80%
We can use raw data
Northeast
D
Southeast
E
Midwest
F
West
G
Finance
A
24
10
8
14
56
Manufacturing
B
30
6
22
12
70
Communication
C
28
18
12
16
74
82
34
42
42
200
Or we can convert our contingency
table into percentages
Northeast
D
Southeast
E
Midwest
F
West
G
Finance
A
.12
.05
.04
.07
.28
Manufacturing
B
.15
.03
.11
.06
.35
Communication
C
.14
.09
.06
.08
.37
.41
.17
.21
.21
1.00
P(Midwest) = ?
P(C or D) = ?
P(E or A) =?
North
east
D
Finan
ce
A
Manuf
acturi
ng B
Com
munic
ation
C
.12
.15
South
east
E
.05
.03
Midw
est
F
.04
.11
North
east
D
South
east
E
Midw
est
F
West
G
.28
Finan
ce
A
24
10
8
14
56
.35
Manuf
acturi
ng B
30
6
22
12
70
Com
munic
ation
C
28
18
12
16
74
82
34
42
42
200
West
G
.07
.06
.14
.09
.06
.08
.37
.41
.17
.21
.21
1.00
Solution
P(F) = .21
P(C or D) =
P(C) + P(D) – P(C & D)
= .37 + .41 - .14
= .64 or 64%
P(E or A) =
.17 + .28 - .05
= .40 or 40%
North
east
D
South
east
E
Midwe
st
F
West
G
Finan
ce
A
.12
.05
.04
.07
.28
Manuf
acturi
ng B
.15
.03
.11
.06
.35
Comm
unicat
ion C
.14
.09
.06
.08
.37
.41
.17
.21
.21
1.00
Addition Rule for Mutually
Exclusive Events:
P(A or B) = P(A) + P(B)
Frequently, we need to know how
two events are related.
Conditional Probability
We would like to know the
probability of one
event occurring given
the occurrence of
another related event.
Conditional Probability
The probability that Event A occurs GIVEN that
Event B occurs.
P (A | B)
B is the event known to have occurred and A is
the uncertain event whose probability you
seek, given that Event B has occurred.
Gender
Male (M)
Female (F)
Total
<30
(U)
60
40
100
Age (Years)
30-45
(B)
20
30
50
>45
(O)
40
10
50
Total
120
80
200
What is the probability of selecting a female purchaser
given the selected individual is under 30 years of age?
P(F | U) = 40 / 100 = .4
Interpretation:
There is a 40% probability of selecting a female given the
selected individual is under 30 years of age.
Hypoxia Question 1:
How is P(F|U)
different than
the P(F)?
There is a 40% chance of selecting a
female purchaser given no prior
information about U. P(F)= .4
This means that being given the
information that the person selected is
under 30 has no effect on the
probability that a female is selected.
In other words, U has no effect on
whether F occurs. Such events are
said to be INDEPENDENT
Events A and B are independent if
the probability of Event A is unaffected
by the occurrence or non-occurence of
Event B
Statistical Independence
•
•
•
•
Events A and B are independent if and
only if:
P(A | B) = P(A) {assuming P(B) ≠ 0}, or
P(B | A) = P(B) {assuming P(A) ≠ 0}, or
P(A and B) = P(A) ∙ P(B).
Gender
Male (M)
Female (F)
Total
<30
(U)
60
40
100
Age (Years)
30-45
(B)
20
30
50
>45
(O)
40
10
50
What is the probability of
selecting a female purchaser
given the selected individual
is between 30-45 years of
age?
Are the events independent?
Total
120
80
200
P(F | B) = 30/50 = .6
Test for independence:
P(F | B) = P(F)
30/50 = 80/200
.6 ≠ .4
The events are not independent.
1) Suppose we have the following joint probabilities.
A1
.15
.25
B1
B2
1)
2)
3)
4)
5)
6)
7)
A2
.20
.25
Compute the marginal probabilities.
Compute P(A2 | B2)
Compute P(B2 | A2)
Compute P(B1 | A2)
Compute P( A1 or A2)
Compute P(A2 | or B2)
Compute P(A3 or B1)
A3
.10
.05
1) The female instructors at a large university recently lodged a complaint about the most recent round of
promotions from assistant professor to associate professor. An analysis of the relationship between gender and
promotion was undertaken with the joint probabilities in the following table being produced.
•
Female
Male
Promoted
.03
.17
Not Promoted
.12
.68
•
•
What is the rate of
promotion among
female assistant
professors?
What is the rate of
promotion among male
assistant professors?
Is it reasonable to
accuse the university
of gender bias?
To determine whether drinking alcoholic beverages has an effect on the bacteria that cause ulcers, researchers
developed the following table of joint probabilities.
Number of alcoholic drinks per
day
None
One
Two
More than two
i)
ii)
iii)
iv)
Ulcer
No Ulcer
.01
.03
.03
.04
.22
.19
.32
.16
What proportion of people have ulcers?
What is the probability that a teetotaler (no alcoholic beverages) develops an ulcer?
What is the probability that someone who has an ulcer does not drink alcohol?
Are ulcers and the drinking of alcohol independent? Explain.