Introduction to Probability
ASW, Chapter 4
Skip section 4.5
September 17 and 22, 2008
Why study probability
Inferential statistics – inferences about population
from:
– Sample
– Experiment (or random experiment – ASW, 149)
Economic models are often probabilistic
Quantifying uncertainty for purposes of:
– Prediction
– Assessing preferred course of action
Definition of probability
Probability is a numerical measure of the
likelihood that an event will occur (ASW, 142).
Notation: The probability of an event E is
written P(E) and pronounced “P of E”
Scale: 0 ≤ P(E) ≤ 1
If P(E) = 0, event E cannot occur
If P(E) = 1, event E is certain to occur
Terms for probability
•
•
•
•
•
•
•
Probability
Likelihood
Odds
Chances
Risk or hazard
Random selection
Random or stochastic processes
Experiments and outcomes
An experiment is a process that generates welldefined outcomes (ASW, 142)
– Tossing a coin
– Rolling a single die or a pair of dice
– Applying for a job
– Salary expected after completing a degree
– Drilling an oil well
– Federal election
ASW note that these are sometimes termed
random experiments (150).
Assigning probabilities
• Classical or theoretical method (ASW, 146)
• Relative frequency method (ASW, 148)
– Data and experiments
• Subjective or judgment method (ASW, 148). A
degree of belief that an outcome will occur
– Best guess / experience / wise judgment produces
good estimates of the chance of a particular
outcome occurring
– Poor guess / poor judgment produces unreliable
estimates of the chance of a particular outcome
occurring
Classical method
Experiment (eg. Flip of a coin – outcome of H or T)
• Exact outcome is unknown before conducting
experiment
• All possible outcomes of experiment are known
• Each outcome is equally likely
• Experiment can be repeated under uniform
conditions
Together these conditions produce regularities or
patterns in outcomes
Examples of classical method
• P (2 heads in two tosses of a coin) = 1/4 = 0.250
• P (obtain 4 in roll of one die) = 1/6 = 0.167
• P (total of 4 when rolling two dice) = 3/36 = 1/12 =
0.083
If one randomly selects individuals from a large
population that is ½ male and ½ female:
• P (one male and one female if 2 individuals selected)
= 2/4 = 1/2 = 0.500
• P (two females and one male if 3 individuals
selected) = 3/8 = 0.375
Relative frequency method
This method used for an experiment where it is
not possible to apply the classical approach
(usually because outcomes not equally likely
or the experiment is not repeatable under
uniform conditions).
The probability of an event E is the relative
frequency of occurrence of E or the
proportion of times E occurs in a large number
of trials of the experiment.
Example of relative frequency method
Party supported Per cent
support
Conservative
Liberal
NDP
38
27
17
Green
9
Bloc Québécois
8
If I meet an individual Canadian
and I know nothing about this
person, the best estimate of the
probabilities that the person
supports the various parties are:
P (individual is a Conservative
supporter) = 0.38
P (individual supports the Green
Party) = 0.09
Source:
http://www.theglobeandmail.com/natio
nal/politics/, 12:15 p.m., Sept. 17,
2008
Subjective or judgment method
• Make your best guess of the likelihood of the event,
based on reliable information and good judgment.
• Inform yourself about the issue or discuss it with
someone knowledgeable.
• Probability is a number between 0 and 1 that
represents your degree of belief the event will occur.
• P(E) close to 0 implies a judgment that the event is
unlikely to occur.
• P(E) close to 1 implies a judgment that the event is
quite likely to occur.
Examples of subjective method –
my subjective judgment
• P (UR Rams will defeat the SFU Clan this
Saturday) = 0.7
• P (S & P TSX index will increase in trading on
Monday) = 0.3
• P (I will have an auto accident during the next
6 months) = 0.02
• P (some snow will fall in Regina before the end
of October) = 0.97
Which method to use?
• Whenever it is possible to apply it, use the
classical method.
• When dealing with economic activity, the classical
method often cannot be used (not equally likely
events and not repeatable). Obtain data and use
the relative frequency method.
• If neither of these is possible, use the subjective
method. It is a weak method, but if judgments
are sound, these subjective probabilities can be
used in coordination with economic theory and
more soundly based probabilities.
Sample space (ASW, 141)
• An experiment is a process that generates
well-defined outcomes.
• The sample space, S, for an experiment is the
set of all experimental outcomes.
• Each of these outcomes is referred to as a
sample point. For an experiment with n
sample points, label these E1, E2, E3, ... , En.
• The sample space of an experiment is the set
of all sample points, ie. S = {E1, E2, E3, ... , En}.
Two requirements for assigning
probabilities
• If Ei is the ith outcome of an experiment, then
its probability is no less than zero and no
greater than 1. That is, for all sample points
or outcomes, Ei, 0 ≤ P(Ei) ≤ 1, for i = 1, 2, ... n.
• The sum of the probabilities of all the
outcomes equals 1, that is,
P(E1) + P(E2) + P(E3) + ... + P(En) = 1 or

i n
i 1
P( Ei )  1
Examples of sample spaces
• Flip a coin (n = 2). S = {H, T}, where H = head, T= tail.
P(H) = 1/2 = 0.50 and P(T) = 1/2 = 0.50.
P(H) + P(T) = 0.50 + 0.50 = 1.
• One roll of a ten-sided die (n = 10). S = {0, 1, 2, 3, 4, 5,
6, 7, 8, 9}.
Probability of each side occurring is 1/10.
P(0) + P(1) + P(2) + ... + P(9) =1.
• Party that will form the government following the
Canadian federal election. S = {C, L, N, G, B} where C =
Conservative, L = Liberal, N = NDP, G = Green, B = Bloc.
What are P(C), P(L), P (N), P(G), P(B)?
P(C) + P(L) + P(N) + P(G) + P(B) = 1.
Multiple-step experiments
• If an experiment with k steps has n1 possible
outcomes on the first step, n2 possible
outcomes on the second step, etc., then the
sample space has n1 × n2 × n3 × ... × nk sample
points (ASW, 143).
• If 3 persons are randomly selected from a
large population with half females and half
males, there are 2 × 2 × 2 = 8 outcomes for the
sex of the persons selected. Each outcome is
equally likely.
Second
selection
First
selection
F
Probability
F
(F, F, F)
1/8 = 0.125
M
F
(F, F, M)
1/8 = 0.125
(F, M, F)
1/8 = 0.125
(F, M, M)
1/8 = 0.125
(M, F, F)
1/8 = 0.125
(M, F, M)
1/8 = 0.125
•
F
•
M
•
F
M
Third
Sample
selection points
•
•
•
Selection of 3 persons
from a large population
of half females and half
males.
M
F
M
F
M
•
M
(M, M, F) 1/8 = 0.125
(M, M, M) 1/8 = 0.125
Sum = 1.000
Event
• An event is a collection of sample points
(ASW, 152).
• The probability of any event is equal to the
sum of the probabilities of the sample points
in the event (ASW, 153).
• Event of selecting exactly two females (2F).
P (2F) = 1/8 + 1/8 + 1/8 = 3/8 = 0.375.
This is the sum of the probabilities of the
events FFM, FMF, and MFF.
Results from Econ 224 survey
• Sample of N = 46 students.
• Step one – results organized into four groups
by major: Math (including Actuarial and
Statistics), Business, Economics, and Other.
• Step two – results organized into four groups
of experience level with Excel – None (N), Low
(L), Medium (M), or High (H).
• Number of students reporting each
characteristic reported in a tree diagram.
Step one:
Major
Step Two:
Excel Skill
N = 46
Outcome:
N=0
Math
(6)
L=2
M=4
H=0
N=1
MAJOR
Business
(13)
L=3
M=6
H=3
N=2
Economics
(24)
L = 12
M=8
H=2
N=0
Other
(3)
L=1
M=1
H=1
Probabilities for Excel skill levels
If each outcome equally likely, P(E) = NE/N where
NE is the number of outcomes in event E and N
is the total number of outcomes.
Probability of selecting a student with low Excel
skills = (2 + 3 + 12 +1)/46 = 18/46 = 0.391
P(High Excel skills) = 6/46 = 0.130
P(No Excel experience) = 3/46 = 0.065
P(Medium Excel experience) = 19/46 = 0.413
Note that 0.065 + 0.391 + 0.413 + 0.130 = 0.999
Relationships of Probability
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Complement of an event
Addition law – intersection and union
Mutually exclusive
Conditional probability
Independence
Multiplication law
Complement of an event
• The complement Ac of an event A is the set of
all sample points that are not in event A.
• P(A) + P(Ac) = 1 or P(Ac) = 1 – P(A).
• Examples: The probability that a student in
this class is not an Economics major (E) is
22/46.
P(E) = 24/46 = 0.522
P(Ec) = 1 – 0.522 = 0.478
P(Excel skills not High) = 1 – 0.130 = 0.870
Combining events (ASW, 157-160)
What is the probability that more than one
event has occurred? If there are two events:
Both events could occur – this is referred to as
the intersection of the two events.
At least one of the events could occur – this is
referred to as the union of the two events.
Neither event occurs.
Union of two events
• The union of events A and B is the event
containing all the sample points of either A or
B, or both (ASW, 157).
• The notation for the union is P(AB).
• Read this as “probability of A union B” or the
“probability of A or B.”
• The probability of A or B is the sum of the
probabilities of all the sample points that are
in either A or B, making sure that none are
counted twice.
Intersection of two events
• The intersection of events A and B is the event
containing only the sample points belonging
to both A and B (ASW, 158).
• The notation for the intersection is P(AB).
• Read this as “probability of A intersection B”
or the “probability of A and B.”
• The probability of A and B is the sum of the
probabilities of all the sample points common
to both A and B.
Addition law (ASW, 158)
P(AB) = P(A) + P(B) – P(AB)
The probability that at least one event occurs is
the probability of one event plus the
probability of the other. But to avoid double
counting, the probability of the intersection of
the two events is subtracted.
Examples
• Probability of a randomly selected student
being an Economics major (E) or having high
Excel skills (H)?
P(EH) = P(E) + P(H) – P(EH)
= 24/46 + 6/46 – 2/46
= 28/46 = 0.609
• Probability of Business (B) or low skills (L)?
P(BL) = P(B) + P(L) – P(BL)
= 28/46 = 0.609
• P(Other or medium) = 0.456
Mutually exclusive events (ASW, 160)
• Two events are mutually exclusive if the
events have no sample points in common.
• If two events A and B are mutually exclusive,
the probability of A and B is zero.
• In this case, the probability of A or B is the
sum of the probability of A and the probability
of B. That is,
P(AB) = 0 if A and B are mutually exclusive.
Then P(AB) = P(A) + P(B)
Example of mutually exclusive events
If 3 persons are randomly selected from a large
population of half females and half males,
what is the probability of selecting 3 females
(A) and at least 2 males (B)?
Event A has sample point (F, F, F) .
Event B has sample points (F, M, F), (M, F, M),
(M, M, F) and (M, M, M) .
Events A and B have no sample points in
common, so P(AB) = 0.
Conditional probabilities (ASW, 162-167)
• A conditional probability refers to the
probability of an event A occurring, given that
another event B has occurred.
• Notation: P(A  B)
• Read this as the “conditional probability of A
given B” or the “probability of A given B.“
• These are especially useful in economic
analysis because probabilities of an event
differ, depending on other events occurring.
Formulae for conditional probabilities
• The probability of A given B is
P( A  B)
P( A B) 
P( B)
• The probability of B given A is
P( A  B)
P( B A) 
P( A)
Number of students by major and
Excel skill level
Major of
student
Excel skill level
None (N)
Low (L)
Total
Medium (M)
High (H)
Math (MA)
0
2
4
0
6
Business (B)
1
3
6
3
13
Economics (E)
2
12
8
2
24
Other (O)
0
1
1
1
3
Total
3
18
19
6
46
This table contains the same data as examined earlier, but
reorganized as a table rather than in a tree diagram.
Examples of conditional probabilities from
student survey
• Probability that each major has low skill level?
P(L  MA) = P(L  MA) / P(MA) = (2/46) / (6/46) = 2/6 = 0.333
P(L  B) = 3 / 13 = 0.231
P(L  E) = 0.500
P(L  O) = 0.333
If a student has a high skill level is Excel, what is the probability
his or her major is Business? Other?
P(B  H) = P(B  H) / P(H) = (3/46) / (6/46) = 3/6 = 0.500
P(O  H) = 0.167
Next day
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Independence (ASW, 166)
Multiplication law (ASW, 166)
Random variables (ASW, 186)
Discrete probability distributions (ASW, 189)
Expected value – mean and variance (ASW,
195)
• Binomial probability distribution (ASW, 199)