pb chap005 powerpoints.ppt

8/14/2019 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)


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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


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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


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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


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


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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


)()(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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