Transcript + P(B) - McGraw Hill Higher Education

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McGraw-Hill/Irwin
5
Chapter
Probability
Random Experiments
Probability
Rules of Probability
Independent Events
Contingency Tables
Counting Rules
Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.
Random Experiments
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.
5-2
Random Experiments
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}
5-3
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.
5-4
Probability
What is Probability?
Three approaches to probability:
Approach
Example
Empirical
There is a 2 percent chance
of twins in a randomlychosen birth.
Classical
There is a 50 % probability
of heads on a coin flip.
Subjective
There is a 75 % chance that England will
adopt the Euro currency by 2010.
5-5
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.
5-6
Rules of Probability
Union of Two Events
(Figure 5.5)
• 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.
5-7
Rules of 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.
 may be read as
“and” since both
events occur. This is
a joint probability.
5-8
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.
5-9
Rules of 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
Special Law of Addition
• In the case of mutually
exclusive events, the
addition law reduces to:
P(A  B) = P(A) + P(B)
5-10
Rules of Probability
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
5-11
Rules of Probability
Odds of an Event
• The odds in favor of event A occurring is
P( A)
P( A)
Odds =

P( A ') 1  P( A)
• The odds against event A occurring is
P( A) 1  P( A)
Odds 

P( A)
P( A)
5-12
Independent and Dependent Events
• Event A is independent of event B if the
conditional probability P(A | B) is the same
as the marginal probability P(A).
• When P(A) ≠ P(A | B), then events A and B
are dependent.
Multiplication Law for Independent Events
5-13
Contingency Table
What is a Contingency Table?
• A contingency table is a cross-tabulation of frequencies
into rows and columns. Example below.
• From the table, one can compute marginal probabilities,
conditional probabilities, and check for independence
between the two variables.
5-14
Counting Rules
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.
5-15
Counting Rules
Permutations
• A permutation is an arrangement in a
particular order of randomly sampled items
from a group (i.e. XYZ is different from ZYX).
Combinations
• A combination is an arrangement of items
chosen at random where the order of the
selected items is not important (i.e., XYZ is
the same as ZYX).
5-16