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Transcript of PB Chap005 PowerPoints.ppt
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Probability
Chapter
5 Random Experiments
Probability
Rules of Probability Independent Events
Contingency Tables
Counting Rules
Copyright 2010 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin
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A random exper imentis an observational
process whose results cannot be known in
advance.
The set of all outcomes(S) is the sample
spacefor the experiment.
A sample space with a countable number ofoutcomes is discrete.
Sample Space
Random Experiments
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An eventis any subset of outcomes in the
sample space.
A simple eventor elementary event, is a
single outcome.
A discrete sample space Sconsists of all the
simple events (Ei):
S= {E1, E2, , En}
Events
Random Experiments
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The probabi l i tyof 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.
Defin i t ions
Probability
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Three approaches to probability:
Approach Example
Empirical There is a 2 percent chance
of twins in a randomly-
chosen birth.
What is Probabi l i ty?
Probability
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.
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The complementof an event A is denoted by
Aand consists of everything in the samplespace Sexcept event A .
Complement o f an Event
Rules of Probability
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The unionof two events consists of all
outcomes in the sample space Sthat arecontained either in event A or in event Bor
both (denoted A Bor A or B).may be read asor since one or
the other orboth
events may occur.
Union o f Two Events(Figure 5.5)
Rules of Probability
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The intersect ionof two events A and B
(denoted A Bor A and B) is the event
consisting of all outcomes in the samplespace Sthat are contained in bothevent A
and event B.
may be read asand since both
events occur. This is
ajoint probability.
Intersect ion of Two Events
Rules of Probability
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The general law o f add it ionstates 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.
So, you have to
subtract
P(AB) to avoidover-stating the
probability.A B
A and B
General Law o f Add i t ion
Rules of Probability
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Events A and Bare mutually exclus ive(or dis jo in t) if
their intersection is the null set () that contains noelements.
If A B= , then P(A B) = 0
In the case of mutually
exclusive events, the
addition law reduces to:
P(A B) = P(A ) + P(B)
Mutual ly Exclus ive Events
Rules of Probability
Special Law of Addi t ion
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The probability of event A giventhat event B
has occurred.
Denoted P(A | B).
The vertical line | is read as given.
( )( | )( )
P A BP A BP B
for P(B) > 0 and undefinedotherwise
Condit ional Probabi l i ty
Rules of Probability
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The oddsin favo r of event A occurring is
The oddsagainstevent A occurring is
( ) ( )
Odds = ( ') 1 ( )
P A P A
P A P A
Odds of an Event
Rules of Probability
)()(1
)()(Odds
AP
AP
AP
AP
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Event A is independent of event Bif the
conditional probability P(A | B) is the same
as the marginal probability P(A ).
Independent and Dependent Events
When P(A ) P(A | B), then events A and B
are dependent.
Mult ip l icat ion Law fo r Independent Events
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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.
Contingency Table
What is a Con t ingency Table?
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If event A can occur in n1ways and event B
can occur inn
2ways, then eventsA
andB
can occur in n1x n2ways.
In general, mevents can occur
n1x n2x x nmways.
Counting Rules
Fundamental Rule of Count ing
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A permutat ionis an arrangement ina
part icular o rderof randomly sampled items
from a group (i.e. XYZis different from ZYX).
Counting Rules
Permutat ions
Combinat ions
A combinat ionis an arrangement of items
chosen at random where the order of the
selected items is not important (i.e., XYZis
the same as ZYX).
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