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Randomness
We call a phenomenon random if
individual outcomes are uncertain but
there is nonetheless a regular distribution
of outcomes in a large number of
repetitions.
This definition is given by David S. Moore and George P. McCabe
in their book, Introduction to the Practice of Statistics.
Probability
The probability of any outcome of a
random phenomenon is the proportion
of times the outcome would occur in a
very long series of repetitions. That is,
the probability of an outcome is its longterm relative frequency.
This definition is given by David S. Moore and George P.
McCabe in their book, Introduction to the Practice of Statistics.
Some Important Definitions
• An experiment is an act or process of observation that
leads to a single outcome that cannot be predicted with
certainty.
• An sample point is the most basic outcome of an
experiment.
• The sample space of an experiment is the collection of
all its sample points. The sample space is denoted by the
symbol, S.
• An event is a specific collection of sample points.
An event is a subset of the sample space. An event is
denoted by a capital letter such as A.
Probability Rules
• The probability of the event A is
denoted P(A).
• For any event A, 0 < P(A) < 1.
• If S is the sample space in a probability
model, P(S) = 1.
• The complement of any event A is the
event A does not occur, written as Ac.
• The complement rule states that
P(Ac) = 1 – P(A).
Probabilities in a Finite
Sample Space
A sample space is finite if it has a finite
(fixed or limited) number of outcomes.
The probability of each outcome is an
number between 0 and 1.
The sum of these probabilities must be 1.
The probability of any event is the sum of
the probabilities of the outcomes making
up that event.
More Definitions
• If an outcome belongs to both the event A
and the event B, it is said to belong to the
event A and B denoted by A _ B.
• If an outcome belongs to event A or to event
B or to both, the outcome is said to belong to
the event A or B denoted by A ^ B.
• Two events A and B are said to be mutually
exclusive if they have no outcomes in
common.
P(A _ B) = 0.
• If the events A and B are mutually exclusive,
then P(A ^ B) = P(A) + P(B).
An Example
• Suppose there are 5 people who work
for a company in management level
positions.
• Jorge is a male assistant manager.
• Maria is a female assistant manager.
• David is a male manager.
• Jim is a male assistant manager.
• Kim is a female manager.
The experiment will be to choose a person
from this group and record the person’s
name, gender and title.
• The sample space is S.
S = {Jorge, Maria, David, Jim, Kim}.
• Let A be the event a male is chosen.
A = {Jorge, David, Jim }.
• Let B be the event an assistant manger
is chosen.
B = {Jorge, Maria, Jim}.
Venn Diagram of the Sample
Space S with Events A and B
Maria
Jorge
Jim
A
David
A = {Jorge, Jim, David}
B = {Jorge, Jim, Maria}
Kim
B
Some Events Related to Events A and B
• Ac is the complement of A.
– Since A is the event a male is chosen, Ac is the
event a male is not chosen.
Ac = {Maria, Kim}.
• A and B is the event both A and B occur.
– A and B is the event a male assistant manager is
chosen.
A _B = {Jorge, Jim }.
• A or B is the event either A or B or both
occur.
– A or B is the event either a male or an assistant
manager is chosen.
A ^ B = {Jorge, Jim, Maria, David}.
Assigning Probabilities to Events
Assume that each person in our group is
equally likely to be chosen, then
P(Jorge) = P(Maria) = P(David) = P(Jim) =
P(Kim) = 1/5.
Using this information we can assign
probabilities to the events A and B.
P(A) = P(Jorge) + P(David) + P(Jim) = 3/5.
More Probabilities
• P(B) = P(Jorge) + P(Maria) + P(Jim) = 3/5.
• P(A _ B) = P(Jorge) + P(Jim) = 2/5.
• P(A ^ B) = P(Jorge) + P(Maria) + P(Jim) +
P(David) = 4/5.
• P(Ac) = P(Maria) + P(Kim) = 2/5, or
P(Ac) = 1- P(A) = 1 –3/5 = 2/5.
How do we describe the event, {Maria,
Kim} using the letters A and B?
Maria
Jorge
Jim
A
David
Kim
Ac = {Maria, Kim}
B
How do we describe the event,
{Kim} using the letters A and B?
Maria
Jorge
Jim
A
David
Kim
Ac _ Bc = {Kim}
B
The Two-way Frequency Table for
Exercise 3.50 in Statistics, 9th Edition
Died from Cancer
Cigars
Yes
(D)
No
(E)
Never Smoked (A)
782
120,747
121,529
Former Smoker (B)
91
7,757
7,848
Current Smoker (C)
141
7,725
7,866
1,014
136,229
137,243
Totals
Totals
In this example, the 137,243 American males are the
sample points. Since we are interested only in the cigar
smoking status and whether the man died of a smoking
related cancer, the cells in this table represent the sample
points. The table tells us that 782 of these men never
smoked but died of smoking related cancers.
The experiment is to choose an American male
from this group and record his cigar smoking
status and whether he died from a smoking
related cancer.
In the table, I have given each row and column a
letter name. These letters represent events.
A is the event the American male chosen never
smoked cigars.
D is the event the American male chosen died
from a smoking related cancer.
What is the probability that a randomly
selected male never smoked cigars?
Died from Cancer
A
Cigars
Yes
(D)
Never Smoked (A)
782
Former Smoker (B)
91
7,757
7,848
Current Smoker (C)
141
7,725
7,866
Totals
1,014
No
(E)
Totals
120,747 121,529
136,229 137,243
782
120 ,747 121,529
+
=
P ( A) =
137 ,243 137 ,243 137 , 243
What is the probability that a randomly
selected male never smoked cigars and
died of a smoking related cancer (A _ B)?
Died from Cancer
A
Cigars
Yes
(D)
Never Smoked (A)
782
Former Smoker (B)
91
7,757
7,848
Current Smoker (C)
141
7,725
7,866
Totals
1,014
No
(E)
Totals
120,747 121,529
136,229 137,243
782
P( A D) =
137 ,243
What is the probability that a randomly
selected male is either a former smoker or
died of a smoking related cancer (B ^D)?
Died from Cancer
B
Cigars
Yes
(D)
Never Smoked (A)
782
Former Smoker (B)
91
7,757
7,848
Current Smoker (C)
141
7,725
7,866
Totals
1,014
No
(E)
Totals
120,747 121,529
136,229 137,243
P(B * D) = P(B) + P(D) − P(B D)
7,848
1,014
91
8,771
=
+
−
=
137 ,243 137 ,243 137 ,243 137 ,243
What is the probability that a randomly selected
male is not a current smoker and did not die of a
smoking related cancer (Cc _ E)?
Died from Cancer
Cigars
Yes
(D)
Never Smoked (A)
782
Former Smoker (B)
91
7,757
7,848
Current Smoker (C)
141
7,725
7,866
Totals
1,014
No
(E)
Totals
120,747 121,529
136,229 137,243
P (C E ) = P ( A E ) + P ( B E )
c
120 ,747
7,757
128 ,504
=
+
=
137 ,243 137 ,243 137 ,243