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Chapter 5
Probability
Chapter Contents
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
Random Experiments
Probability
Rules of Probability
Independent Events
Contingency Tables
Tree Diagrams
Bayes’ Theorem
Counting Rules
5-2
Chapter 5
Probability
Chapter Learning Objectives
LO5-1:
LO5-2:
LO5-3:
LO5-4:
LO5-5:
LO5-6:
LO5-7:
LO5-8:
LO5-9:
Describe the sample space of a random variable.
Distinguish among the three views of probability.
Apply the definitions and rules of probability.
Calculate odds from given probabilities.
Determine when events are independent.
Apply the concepts of probability to contingency tables.
Interpret a tree diagram.
Use Bayes’ Theorem to calculate revised probabilities.
Apply counting rules to calculate possible event arrangements.
5-3
Chapter 5
5.1 Random Experiments
LO5-1
LO5-1: Describe the sample space of a random experiment.
Sample Space
•
•
•
A random experiment is an observational process whose results cannot be known in
advance.
The set of all outcomes (S) is the sample space for the experiment.
A sample space with a countable number of outcomes is discrete.
Events
•
•
•
An event is any subset of outcomes in the sample space.
A simple event or elementary event, is a single outcome.
A discrete sample space S consists of all the simple events (Ei): S = {E1, E2, …, En}
5-4
Chapter 5
LO5-2
5.2 Probability
LO5-2: Distinguish among the three views of probability.
Definitions
•
The probability of an event is a number that measures the relative likelihood that the
event will occur.
•
The probability of event A [denoted P(A)], must lie within the interval from 0 to 1:
0 < P(A) < 1
If P(A) = 0, then the event
cannot occur.
If P(A) = 1, then the event
is certain to occur.
5-5
Chapter 5
LO5-1
5.2 Probability
Law of Large Numbers
•
The law of large numbers is an important probability theorem that states
that a large sample is preferred to a small one.
5-6
Chapter 5
LO5-3
5.3 Rules of Probability
LO5-3: Apply the definitions and rules of probability.
Complement of an Event
•
The complement of an event A is denoted by
A′ and consists of everything in the sample space S except event A.
•
Since A and A′ together comprise
the entire sample space,
P(A) + P(A′ ) = 1 or P(A′ ) = 1 – P(A)
5-7
Chapter 5
LO5-3
5.3 Rules of Probability
Union of Two Events
(•
The union of two events consists of all outcomes in the sample space S that are
contained either in event A or in event B or both (denoted A  B or “A or B”).
 may be read as “or”
since one or the other or
both events may occur.
 may be read as “and”
since both events occur.
This is a joint probability.
Intersection of Two Events
•
The intersection of two events A and B
(denoted A  B or “A and B”) is the
event consisting of all outcomes in the
sample space S that are contained in
both event A and event B.
5-8
Chapter 5
LO5-3
5.3 Rules of Probability
General Law of Addition
•
The general law of addition states that the probability of the union of two events A
and B is:
P(A  B) = P(A) + P(B) – P(A  B)
When you add the P(A)
and P(B) together, you
count the P(A and B)
twice.
A and B
A
B
So, you have to
subtract P(A  B) to
avoid over-stating the
probability.
Mutually Exclusive Events
•
Events A and B are mutually
exclusive (or disjoint) if their
intersection is the null set ()
that contains no elements.
If A  B = , then P(A  B) = 0
5-9
Chapter 5
LO5-3
5.3 Rules of Probability
Special Law of Addition
•
In the case of mutually exclusive
events, the addition law reduces to:
P(A  B) = P(A) + P(B)
Conditional Probability
•
•
The probability of event A given that event B has occurred.
Denoted P(A | B). The vertical line “ | ” is read as “given.”
P( A  B)
P( A | B) 
P( B)
for P(B) > 0 and undefined otherwise
5-10
Chapter 5
LO5-4
5.3 Rules of Probability
LO5-4: Calculate odds from given probabilities.
Odds of an Event
•
The odds in favor of event A occurring is
P( A)
P( A)
Odds =

P( A ') 1  P( A)
•
•
If the odds against event A are
quoted as b to a, then the implied
probability of event A is:
The odds against event A occurring is
P( A) 1  P( A)
Odds 

P( A)
P( A)
5-11
Chapter 5
LO5-5
5.4 Independent Events
LO5-5: Determine when events are independent.
•
•
•
Event A is independent of event B if the conditional probability P(A | B) is the same
as the marginal probability P(A).
Another way to check for independence:
If P(A  B) = P(A)P(B) then event A is independent of event B since
P(A | B) = P(A  B) = P(A)P(B) = P(A)
P(B)
P(B)
5-12
Chapter 5
LO5-6
5.5 Contingency Table
LO5-6: Apply the concepts of probability to contingency tables.
•
Example:
P(T1 and S1) = 5/67 P(T2 and S2) = 11/67
P(S1) = 17/67
P(T3 and S3) = 15/67
P(T2) = 19/67
5-13
Chapter 5
LO5-7
5.6 Tree Diagrams
LO7: Interpret a tree diagram.
What is a Tree?
•
A tree diagram or decision tree helps you visualize all possible outcomes.
•
A tree diagram shows all events along with their marginal, conditional and joint
probabilities. Example:
5-14
Chapter 5
LO5-8
5.7 Bayes’ Theorem
LO5-8: Use Bayes’ Theorem to compute revised probabilities.
•
Thomas Bayes (1702-1761) provided a method (called Bayes’ Theorem) of revising
probabilities to reflect new probabilities.
•
The prior (marginal) probability of an event B is revised after event A has been
considered to yield a posterior (conditional) probability.
•
Bayes’ formula is:
•
In some situations P(A) is not given. Therefore, the most useful and common form
of Bayes’ Theorem is:
P( B | A) 
P( B | A) 
P( A | B) P( B)
P( A)
P( A | B) P( B)
P( A | B) P( B)  P( A | B ') P( B ')
5-15
Chapter 5
LO5-8
5.7 Bayes’ Theorem
General Form of Bayes’ Theorem
• A generalization of Bayes’ Theorem allows event B to have as many mutually
exclusive and collectively exhaustive categories as we wish (B1, B2, … Bn) rather than
just two dichotomous categories (B and B').
P( A | Bi ) P( Bi )
P( Bi | A) 
P( A | B1 ) P( B1 )  P( A | B2 ) P( B2 )  ...  P( A | Bn ) P( Bn )
LO5-9: Apply counting rules to calculate possible event arrangements.
Fundamental Rule of Counting
•
•
If event A can occur in n1 ways and event B can occur in n2 ways, then events A and
B can occur in n1 x n2 ways.
In general, m events can occur n1 x n2 x … x nm ways.
5-16
Chapter 5
LO5-9
5.8 Counting Rules
Permutations
•
A permutation is an arrangement in a particular order of r randomly sampled items from
a group of n items and is computed from:
Combinations
•
A combination is an arrangement of r items chosen at random from n items where
the order of the selected items is not important and is computed from:
Note: n! = n(n–1)(n–2)...1, etc.
5-17