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Probability and Probability
Distributions
ASW, Chapter 4-5
Skip sections 4.5, 5.5, 5.6
September 24, 2008
Conditional probabilities (ASW, 162167)
• 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.”
• Conditional probabilities are especially useful in
economic analysis because probabilities of an
event differ, depending on other events
occurring.
Formulae for conditional
probabilities
• The conditional probability of A given B is
P( A  B)
P( A B) 
P( B)
• The conditional 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
Using conditional probabilities
• “While four-day workweeks make some sense for the
manufacturing sector, it’s much more challenging for
service-based companies that have to be available for
clients’ questions. Even on Fridays.”
Source: The Globe and Mail, September 20, 2008, B18.
• Parents who resided in the largest census metropolitan
areas were more likely to have an adult child at home.
For example, 41% of parent in Vancouver but only 17%
of parent living in rural areas or small towns shared their
house with at least one adult child.
Source: “Parents with adult children living at home.” Statistics Canada, Canadian
Social Trends, Spring 2006.
Sample of Saskatchewan residents
• Random sample of 2,500 Saskatchewan
residents from the Census of Canada, 2001
Public Use Microdata File, Individuals File.
Obtained from the Internet Data Library System,
through the University of Regina Data Library
Services.
• Subgroup selected was those with ages 30-64
years, wages and salaries greater than zero,
and full-time jobs in the year 2000.
• This resulted in a sample of 700 individuals.
Number of Saskatchewan residents with various
levels of wages and salaries and schooling, 2000
Wages and
salaries
Years of schooling
<12
12-13
14-17
Total
18+
<$20,000
38
84
47
8
177
$20-45,000
69
135
101
20
325
$45,000+
21
72
82
23
198
128
291
230
51
700
Total
Some conditional probabilities
What is the conditional probability of $45,000 in wages and
salaries given less than twelve years of schooling?
Given 14-17 years of schooling? Given 18+ years?
P(45+  <12) = P(45+  <12) / P(<12) = 21/ 128 = 0.164
P(45+  14-17) = 82/ 230 = 0.357
P(45+  18+) = 0.451
That is, chances of a high income increase with each
higher level of schooling.
What is the probability that someone with a middle level of
income has 12-13 years of schooling?
P(12-13  20-45) = P(12-1320-45) / P(<20-45) = 135/ 325
= 0.415
Conditional probabilities of various levels of wages
and salaries, given years of schooling, n=700
Saskatchewan residents, 2000
Wages and
salaries
<$20,000
Years of schooling
<12
12-13
14-17
Total
18+
0.297 0.289 0.204 0.157 0.253
$20-45,000 0.539 0.464 0.439 0.392 0.464
$45,000+
0.164 0.247 0.357 0.451 0.283
1.000 1.000 1.000 1.000 1.000
Organizing cross-classification tables
ASW use joint probability tables with joint and marginal
probabilities. Study the example on pages 163-164.
– Joint probabilities are the probabilities of the
intersection of each pair of events in a crossclassification table.
– Marginal probabilities are the probabilities of each of
the events in the rows and columns of the table.
Conditional probabilities can be computed from the
numbers of cases, as reported in the cross-classification
table, as in the examples shown above. I find this
method more useful for the following analysis of
independence and dependence.
Independent and dependent events
(ASW, 166)
Two events A and B are independent if
P(A  B) = P(A) or P(B  A) = P(B).
That is, the probability of one event is not altered by
whether or not the other event occurs.
If P(A  B) = P(A), then P(B  A) = P(B), and vice-versa.
Two events A and B are dependent if
P(A  B) ≠ P(A) or P(B  A) ≠ P(B).
In this case, the occurrence of one event affects the
probability of the other event.
Example of dependence
Does the event of having low wages and salaries
depend on having few years of schooling?
If A is the event of having a low salary (<$20,000) and B is
the event of having less than twelve years of schooling
P(A  B) = 38/128 = 0.297
P(A) = 177/700 = 0.253
And P(A  B) > P(A) so the chance of having low wages and
salaries is greater for those with the least amount of
schooling, as compared with the whole sample.
Also note in this case that P(B  A) = 38/177 = 0.215 >
0.183 = P(B). This is an alternative way of checking for
whether the events are dependent or independent.
Example of independence
Are the events of having 12-13 years of schooling (A) and
the event of having wages and salaries of $20-45,000
(B) dependent or independent?
P(A  B) = 135/325 = 0.415
P(A) = 291/700 = 0.416
So these two events are essentially independent of each
other. Also note that
P(B  A) = 135/291 = 0.464
P(B) = 325/700 = 0.464
In this case, those with a middle level of schooling (12-13
years) and the middle category of income are similar to a
cross-section of the whole sample.
Using dependence and independence
Some authors have argued that parents in higher socioeconomic positions may have a greater tendency to
expect their children to be independent earlier than those
with less education and income….However, the
analysis…does not show support for these
interpretations. Parents with a higher level of education
were neither more not less likely than less well-educated
parents to live with their adult children. Nor were
parents with high personal income any less likely than
those with lower personal income to provide
accommodation for their children.
Source: “Parents with adult children living at home.” Statistics Canada, Canadian
Social Trends, Spring 2006.
Independence and dependence in
economic analysis
Is the price of wheat received by
Saskatchewan farmers dependent on the
weather in Russia?
Is the chance of NAFTA being renegotiated
dependent on the result of the U.S.
presidential election?
Is the consumption of table salt dependent
on interest rates? Is it dependent on
health fads?
Multiplication rule (ASW, 165)
The multiplication rule can be used to compute the
probability of the intersection of two events.
P(A ∩ B) = P(A) P(B  A)
P(A ∩ B) = P(B) P(A  B)
But note that if events A and B are independent of
each other, then P(B  A) = P(B) and P(A  B) =
P(A), so that
P(A ∩ B) = P(A) P(B)
Example of multiplication rule
What is the probability of wages and salaries of $20-45,000
(A) and having 12-13 years of schooling (B)?
Since we already know that A and B are independent,
P(A) x P(B) = (325/700) x (291/700) = 0.193.
Note that P(A ∩ B) = 135 / 700 = 0.193 from the table.
What is the probability of wages and salaries of $45,000+
(C) and having 14-17 years of schooling (D)?
In this case, we have not checked for independence, so
use the full formula:
P(C ∩ D) = P(C) P(D  C) = (198/700)X(82/198) = 82/700
= 0.117
Using independence
• Independent trials of an experiment:
– Successive flips of a coin.
– Many rolls of a die or a pair of dice.
– Sale of a product to customers arriving at a retail store.
• If a population is small and a case that is selected is not
replaced before the next case is drawn, then successive
drawings are dependent on each other. But if the
population is large, successive draws do not alter the
composition of the population. Thus, random selection
of respondents from a large population produces
independence of successive selections.
When trials of an experiment are independent of each
other, then the binomial probability distribution can be
used to determine the probability of several occurrences
of an event in many trials– ASW, section 5.4.
Random variables (ASW, 185)
• A random variable is a numerical description of the
outcome of an experiment.
• Or, a random variable attaches a numerical value to
each possible experimental outcome.
• A random variable is often assigned an algebraic symbol
such as x.
• A random variable can be either discrete (countable
number of possible values) or continuous (not countable
or any numerical value with an interval).
• Chapter 5 deals with discrete random variables.
• Chapter 6 deals with continuous random variables.
Discrete random variables
• Any random variable that has a finite number of possible
values or a countably infinite number of possible values.
• Examples:
– The number of females in a sample of 3 persons selected from a
large population that is half female and half male (x = 0, 1, 2, 3).
– The sum of the faces shown when a pair of dice is rolled (x = 2,
3, 4, … , 12).
– The number of customers at a restaurant at lunch (x = 0, 1, 2, 3,
4, 5, … , 45). To the maximum of the number of seats.
– The number of unemployed workers in Saskatchewan reported
by Statistics Canada each month (x = 0, 1, 2, … , 29,800).
– The number of homeowners who have defaulted on mortgages
in the United States during the last year.
Continuous random variables
• Any random variable whose possible values cannot be
counted is termed continuous. Alternatively, if the
possible outcomes can take on any numerical value in
an interval or set of intervals, the random variable is
continuous.
• Examples:
– Number of kilometres goods are transported from a
manufacturing plant to a warehouse.
– Time taken to ship the goods.
– Exchange rates for currencies.
– Household income.
• We will study the continuous uniform distribution and the
normal distribution (bell curve) next week.
Probability distributions
• A probability distribution is a random
variable, along with the associated
probabilities of occurrence of the values of
the variable.
– Discrete – probabilities of each value of the
random variable.
– Continuous – probability that the random
variable is within a particular interval.
Discrete probability distribution. (ASW,
189-192)
For a discrete random variable x, the probability
distribution is the set of values of x, along with
f(x), the function that gives the probability for
each value of x.
For each value of x, f(x) is no less than 0 and no
greater than 1. The sum of the probabilities for
all values of x equals 1. Symbolically,
0  f(x)  1
∑ f(x) = 1
Probability distribution for sex of person selected
Equally likely
outcomes for
experiment of
randomly selecting 3
persons from a large
population of half
males and half
females:
FFF
FFM
FMF
FMM
MFF
MFM
MMF
MMM
Let the random variable x be the
number of females selected and f(x)
the probabilities for each value of x.
x
f(x)
0
1/8 = 0.125
1
3/8 = 0.375
2
3/8 = 0.375
3
1/8 = 0.125
Total
8/8 = 1.000
In this example, the values of
f(x) are obtained using the
classical interpretation of
probability.
Responses to “Would you like to lower tuition,
even if it meant larger class sizes?
Response
Numerical
value
Number of
respondents
Strong no
1
2
0.044
Weak no
2
5
0.111
Indifferent
3
10
0.222
Weak yes
4
8
0.178
Strong yes
5
20
0.444
45
0.999
Total
Relative
frequency
Probability distribution and expected value for
lower tuition question
If a student is
randomly selected, let
x be the response to
the lower tuition
question. In this case,
the values of the
probability function f(x)
are the relative
frequencies of
occurrence of the
responses to the
question.
x
f(x)
xf(x)
1
0.044
0.044
2
0.111
0.222
3
0.222
0.666
4
0.178
0.712
5
0.445
2.225
Total
1.000
E(x) = 3.869
Graphing discrete probability
distributions
• Use a line chart as in Figure 5.1 of ASW. Or it
could be a bar chart with spaces left between
the bars, to visually indicate that it is a discrete
distribution.
• Convention is to place the values of the random
variable x on the horizontal axis and values of
the probability function f(x) on the vertical axis.
• Examples that follow illustrate these methods.
Probability of Statistics Courses Completed
0.50
0.45
0.40
Probability
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0
1
2
3
4
5
Number of Courses Completed
Source: Fall 2005 Survey, prepared by Harvey King
6
7
Expected values (ASW, 195)
The expected value E(x) of a random variable x is the
mean of the probability distribution. Symbolically,
E(x) = μ = ∑ x f(x)
where μ (pronounced something like “mu”) is a Greek
symbol used to indicate mean.
The concept of expected value is more general than just
referring to the mean, in that the expected value can be
obtained for other expressions – see later notes on the
variance. However, in this course, it will be used to
denote the expected value of x, or the mean.
Expected value for x, number of females selected
x
f(x)
x f(x)
0
1/8 = 0.125
0.000
1
3/8 = 0.375
0.375
2
3/8 = 0.375
0.750
3
1/8 = 0.125
0.375
Total
8/8 = 1.000
1.500
E(x) = μ = ∑ x f(x) = 1.500
If a random sample of 3 persons is obtained from a large population
composed of half females and half males, the expected number of
females selected is 1.5. If there are many samples of 3 persons each
time, the mean number of females across the samples is 1.5.
Expected value for responses to lower tuition
question
The expected value of
the responses is
3.869, or 3.9. Recall
that a response of 3
was “indifferent” and a
response of 4 was
“weak yes” so, in this
sample, the expected
value or mean is just
below “weak yes.”
x
f(x)
xf(x)
1
0.044
0.044
2
0.111
0.222
3
0.222
0.666
4
0.178
0.712
5
0.445
2.225
Total
1.000
E(x) = 3.869
Variance (ASW, 195)
The variance of a probability distribution is the expected
value of the squares of the differences of the random
variable x from the mean μ. Symbolically,
Var(x) = σ2 = ∑(x – μ)2 f(x)
The Greek symbol σ is “sigma.”
The variance can be difficult to calculate and interpret. It is
in units that are the square of the random variable x.
Partly because of this, in statistical work it is more
common to use the square root of the variance or σ.
The standard deviation has the same units as x.
Variance of x, number of females selected
x
f(x)
x f(x)
x-μ
(x – μ)2
(x – μ)2f(x)
0
1/8 = 0.125
0.000
-1.5
2.25
0.28125
1
3/8 = 0.375
0.375
-0.5
0.25
0.09375
2
3/8 = 0.375
0.750
0.5
0.25
0.09375
3
1/8 = 0.125
0.375
1.5
2.25
0.28125
Total
8/8 = 1.000
1.500
0.75000
If a random sample of 3 persons is obtained from a large population
composed of half females and half males, the expected number of
females selected is μ = 1.5. The variance of the number of females
selected is Var(x) = σ2 = ∑(x – μ)2 f(x) = 0.75. The standard deviation
is the square root of 0.75, so that σ = 0.866.
Later this class or next day
• Binomial probability distribution
• Continuous probability distributions
• Bring along copies of the Normal
Distribution for Monday and Wednesday,
Sept. 29 and October 1. This is Table 1 of
Appendix B of ASW.