Introduction to Probability

1
Introduction to Probability

• ASW, Chapter 4
• Skip section 4.5

September 17 and 22, 2008
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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

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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

4
Terms for probability

• Probability
• Likelihood
• Odds
• Chances
• Risk or hazard
• Random selection
• Random or stochastic processes

5
Experiments and outcomes

• An experiment is a process that generates
  well-defined 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).

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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

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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

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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

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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.

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Example of relative frequency method
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/national/politics/,
1215 p.m., Sept. 17, 2008
Party supported Per cent support
Conservative 38
Liberal 27
NDP 17
Green 9
Bloc Québécois 8
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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.

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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

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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.

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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.

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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

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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.

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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.

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Third selection
Sample points
Probability
Second selection
(F, F, F)
1/8 0.125
F
First selection
F
(F, F, M)
1/8 0.125

M
F
F
(F, M, F)
1/8 0.125
M

(F, M, M)
M
1/8 0.125
F
(M, F, F)
1/8 0.125

F
(M, F, M)
M
M
1/8 0.125

F
(M, M, F)
1/8 0.125
M
Selection of 3 persons from a large population of
half females and half males.
(M, M, M)
1/8 0.125
M
Sum 1.000
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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.

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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.

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Step one Major Step Two Excel Skill Outcome
Step one Major Step Two Excel Skill Outcome
N 46 Math (6) N 0
Math (6) L 2
Math (6) M 4
Math (6) H 0
MAJOR Business (13) N 1
MAJOR Business (13) L 3
MAJOR Business (13) M 6
MAJOR Business (13) H 3
Economics (24) N 2
Economics (24) L 12
Economics (24) M 8
Economics (24) H 2
Other (3) N 0
Other (3) L 1
Other (3) M 1
Other (3) H 1
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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

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Relationships of Probability

• Complement of an event
• Addition law intersection and union
• Mutually exclusive
• Conditional probability
• Independence
• Multiplication law

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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

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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.

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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.

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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.

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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.

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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

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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)

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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.

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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.

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Formulae for conditional probabilities

• The probability of A given B is
• The probability of B given A is

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Number of students by major and Excel skill level
Major of student Excel skill level Excel skill level Excel skill level Excel skill level Total
Major of student None (N) Low (L) Medium (M) High (H) Total
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.
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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

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Next day

• 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)