doane chapter 05a
Post on 03-Apr-2018
242 Views
Preview:
TRANSCRIPT
-
7/28/2019 Doane Chapter 05A
1/58
-
7/28/2019 Doane Chapter 05A
2/58
Probability (Part 1)
Random ExperimentsProbability
Rules of Probability
Independent Events
Chapter
5
-
7/28/2019 Doane Chapter 05A
3/58
A random experimentis 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.
Sample Space
Random Experiments
-
7/28/2019 Doane Chapter 05A
4/58
For example, when CitiBank makes a consumer
loan, the sample space is:
S = {default, no default}
The sample space describing a Wal-Mart
customers payment method is:
S = {cash, debit card, credit card, check}
Sample Space
Random Experiments
-
7/28/2019 Doane Chapter 05A
5/58
For a single roll of a die, the sample space is:
S = {1, 2, 3, 4, 5, 6}
When two dice are rolled, the sample space isthe following pairs:
Sample Space
Random Experiments
{(1,1), (1,2), (1,3), (1,4), (1,5), (1,6),
(2,1), (2,2), (2,3), (2,4), (2,5), (2,6),
(3,1), (3,2), (3,3), (3,4), (3,5), (3,6),(4,1), (4,2), (4,3), (4,4), (4,5), (4,6),
(5,1), (5,2), (5,3), (5,4), (5,5), (5,6),
(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
S =
-
7/28/2019 Doane Chapter 05A
6/58
Consider the sample space to describe a randomly
chosen United Airlines employee by
2 genders,21 job classifications,
6 home bases (major hubs) and
4 education levels
It would be impractical to enumerate this sample
space.
There are: 2 x 21 x 6 x 4 = 1008 possible outcomes
Sample Space
Random Experiments
-
7/28/2019 Doane Chapter 05A
7/58
If the outcome is a continuous measurement, the
sample space can be described by a rule.
For example, the sample space for the length of arandomly chosen cell phone call would be
S = {allXsuch thatX> 0}
The sample space to describe a randomly chosen
students GPA would be
S = {X| 0.00 0}
Sample Space
Random Experiments
-
7/28/2019 Doane Chapter 05A
8/58
An eventis any subset of outcomes in the sample
space.
A simple eventorelementary event, is a singleoutcome.
A discrete sample space S consists of all the
simple events (Ei):
S = {E1, E2, , En}
Events
Random Experiments
-
7/28/2019 Doane Chapter 05A
9/58
What are the chances of observing a H or T?
These two elementary events are equally likely.
S = {H, T}
Consider the random experiment of tossing a
balanced coin.
What is the sample space?
When you buy a lottery ticket, the sample space
S = {win, lose} has only two events.
Events
Random Experiments
Are these two events equally likely to occur?
-
7/28/2019 Doane Chapter 05A
10/58
For example, in a sample space of 6 simpleevents, we could define the compound events
A compound eventconsists of two or more simple
events.
These are
displayed in a
Venn diagram:
A = {E1, E2}
B = {E3, E5, E6}
Events
Random Experiments
-
7/28/2019 Doane Chapter 05A
11/58
Many different compound events could be defined.
Compound events can be described by a rule.
S = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}
For example, the compound event
A= rolling a seven on a roll of two
dice consists of 6 simple events:
Events
Random Experiments
-
7/28/2019 Doane Chapter 05A
12/58
Theprobabilityof an event is a number that
measures the relative likelihood that the event will
occur. The probability of eventA [denoted P(A)], must lie
within the interval from 0 to 1:
0 < P(A) < 1
IfP(A) = 0, then the
event cannot occur.
IfP(A) = 1, then the event
is certain to occur.
Defini t ions
Probability
-
7/28/2019 Doane Chapter 05A
13/58
In a discrete sample space, the probabilities of all
simple events must sum to unity:
For example, if the following number of purchases
were made by
P(S) = P(E1) + P(E2) + + P(En) = 1
credit card: 32%
debit card: 20%
cash: 35%
check: 18%
Sum = 100%
Defini t ions
Probability
P(credit card) = .32
P(debit card) = .20
P(cash) = .35
P(check) = .18
Sum = 1 0
Probability
-
7/28/2019 Doane Chapter 05A
14/58
Businesses want to be able to quantify theuncertaintyof future events.
For example, what are the chances that next
months revenue will exceed last years average?
How can we increase the chance of positive future
events and decrease the chance of negative future
events? The study ofprobabilityhelps us understand and
quantify the uncertainty surrounding the future.
Probability
-
7/28/2019 Doane Chapter 05A
15/58
Three approaches to probability:
Approach Example
Empirical There is a 2 percent chance
of twins in a randomly-
chosen birth.
What is Probabi li 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.
-
7/28/2019 Doane Chapter 05A
16/58
Use the empiricalorrelative frequencyapproach
to assign probabilities by counting the frequency
(fi) of observed outcomes defined on theexperimental sample space.
For example, to estimate the default rate on
student loans:
P(a student defaults) = f/n
Emp irical Approach
Probability
number of defaults
number of loans=
-
7/28/2019 Doane Chapter 05A
17/58
Necessary when there is no prior knowledge of
events.
As the number of observations (n) increases or the
number of times the experiment is performed, the
estimate will become more accurate.
Emp irical Approach
Probability
-
7/28/2019 Doane Chapter 05A
18/58
The law of large numbers is an important
probability theorem that states that a large sample
is preferred to a small one. Flip a coin 50 times. We would expect the
proportion of heads to be near .50.
A large n may be needed to get close to .50.
However, in a small finite sample, any ratio can be
obtained (e.g., 1/3, 7/13, 10/22, 28/50, etc.).
Law o f Large Numbers
Probability
Consider the results of 10, 20, 50, and 500 coin
flips
-
7/28/2019 Doane Chapter 05A
19/58
Probability
-
7/28/2019 Doane Chapter 05A
20/58
Actuarial science is a high-paying career that
involves estimating empirical probabilities.
For example, actuaries
- calculate payout rates on life insurance,
pension plans, and health care plans
- create tables that guide IRA withdrawalrates for individuals from age 70 to 99
Pract ical Issues for Actuaries
Probability
-
7/28/2019 Doane Chapter 05A
21/58
Challenges that actuaries face:
- Is nlarge enough to say thatf/n has become a
good approximation to P(A)?- Was the experiment repeated identically?
- Is the underlying process invariant over time?
- Do nonstatistical factors override datacollection?
- What if repeated trials are impossible?
Pract ical Issues for Actuaries
Probability
-
7/28/2019 Doane Chapter 05A
22/58
In this approach, we envision the entire sample
space as a collection of equally likely outcomes.
Instead of performing the experiment, we can usededuction to determine P(A).
a priorirefers to the process of assigning
probabilities before the event is observed.
a priori probabilities are based on logic, not
experience.
Class ical Approach
Probability
-
7/28/2019 Doane Chapter 05A
23/58
For example, the two dice experiment has 36
equally likely simple events. The P(7) is
The probability is
obtained a priori
using the classicalapproach as shown
in this Venn diagram
for 2 dice:
number of outcomes with 7 dots 6( ) 0.1667
number of outcomes in sample space 36P A
Class ical Approach
Probability
-
7/28/2019 Doane Chapter 05A
24/58
A subjective probability reflects someones
personal belief about the likelihood of an event.
Used when there is no repeatable randomexperiment.
For example,
- What is the probability that a new truck
product program will show a return oninvestment of at least 10 percent?
- What is the probability that the price of GM
stock will rise within the next 30 days?
Subject ive App roach
Probability
-
7/28/2019 Doane Chapter 05A
25/58
These probabilities rely on personal judgment or
expert opinion.
Judgment is based on experience with similar
events and knowledge of the underlying causal
processes.
Subject ive App roach
Probability
-
7/28/2019 Doane Chapter 05A
26/58
The complementof an eventA is denoted by
A and consists of everything in the sample space
S except eventA.
Complement o f an Event
Rules of Probability
-
7/28/2019 Doane Chapter 05A
27/58
SinceA andA together comprise the entire
sample space,
P(A) + P(A ) = 1
The probability ofA is found by
P(A ) = 1P(A)
For example, The Wall Street Journalreports that
about 33% of all new small businesses fail withinthe first 2 years. The probability that a new small
business will survive is:
P(survival) = 1P(failure) = 1 .33 = .67 or 67%
Complement o f an Event
Rules of Probability
-
7/28/2019 Doane Chapter 05A
28/58
The odds in favor of eventA occurring is
Odds are used in sports and games of chance.
For a pair of fair dice, P(7) = 6/36 (or 1/6).
What are the odds in favor of rolling a 7?
( ) ( )Odds =
( ') 1 ( )
P A P A
P A P A
(rolling seven) 1/ 6 1/ 6 1Odds =
1 (rolling seven) 1 1/ 6 5/ 6 5
P
P
Odds of an Event
Rules of Probability
-
7/28/2019 Doane Chapter 05A
29/58
On the average, for every time a 7 is rolled, there
will be 5 times that it is not rolled.
In other words, the odds are 1 to 5 in favorofrolling a 7.
The odds are 5 to 1 againstrolling a 7.
Odds of an Event
Rules of Probability
In horse racing and other sports, odds are usuallyquoted againstwinning.
-
7/28/2019 Doane Chapter 05A
30/58
If the odds against eventA are quoted as b to a,
then the implied probability of eventA is:
For example, if a race horse has a 4 to 1 odds
againstwinning, the P(win) is
P(A) =a
a b
Odds of an Event
Rules of Probability
1 10.20
4 1 5
a
a b
P(win) = or 20%
-
7/28/2019 Doane Chapter 05A
31/58
The union of two events consists of all outcomes in
the sample space S that are contained either in
eventA or in event B or both
(denotedA Bor A orB).
may be readas or since
one orthe other
orboth events
may occur.
Union of Two Events
Rules of Probability
-
7/28/2019 Doane Chapter 05A
32/58
For example, randomly choose a card from a deck
of 52 playing cards.
It is the possibility of drawing
eithera queen (4 ways)
ora red card (26 ways)
orboth (2 ways).
IfQ is the event that we draw aqueen and Ris the event that we
draw a red card, what is Q R?
Union of Two Events
Rules of Probability
-
7/28/2019 Doane Chapter 05A
33/58
The intersection of two eventsA and B
(denotedA Bor A and B) is the eventconsisting of all outcomes in the sample space S
that are contained in both eventA and event B.
may be readas and since
both eventsoccur. This is a
joint probability.
Intersect ion o f Two Events
Rules of Probability
-
7/28/2019 Doane Chapter 05A
34/58
It is the possibility of gettingboth a queen anda red card
(2 ways).
IfQ is the event that we draw aqueen and Ris the event that we
draw a red card, what is
Q R?
For example, randomly choose a card from a deck
of 52 playing cards.
Intersect ion o f Two Events
Rules of Probability
-
7/28/2019 Doane Chapter 05A
35/58
The general law of addition states that the
probability of the union of two eventsA 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 theP(A and B)
twice.
So, you have
to subtract
P(A B) toavoid over-stating the
probability.
A B
A and B
General Law of Addi t ion
Rules of Probability
-
7/28/2019 Doane Chapter 05A
36/58
For the card example:
P(Q) = 4/52 (4 queens in a deck)
= 4/52 + 26/52 2/52
P(Q R) = P(Q) + P(R)P(Q Q)
Q
4/52
R
26/52
Q and R= 2/52
General Law of Addi t ion
Rules of Probability
= 28/52 = .5385 or 53.85%
P(R) = 26/52 (26 red cards in a deck)P(Q R) = 2/52 (2 red queens in a deck)
-
7/28/2019 Doane Chapter 05A
37/58
EventsA and B are mutually exclusive (ordisjoint)
if their intersection is the null set () that contains
no elements. IfA B = , then P(A B) = 0
In the case of mutually
exclusive events, the
addition law reducesto:
P(A B) = P(A) + P(B)
Mutually Exclus ive Events
Rules of Probability
Special Law of Addi t ion
l f b b l
-
7/28/2019 Doane Chapter 05A
38/58
Events are collectively exhaustive if their union is
the entire sample space S.
Two mutually exclusive, collectively exhaustive
events are dichotomous (orbinary) events.
For example, a car repair
is either covered by thewarranty (A) or not (B).
WarrantyNo
Warranty
Col lect ively Exhaus t ive Events
Rules of Probability
l f b b l
-
7/28/2019 Doane Chapter 05A
39/58
More than two mutually exclusive, collectively
exhaustive events arepolytomous events.
For example, a Wal-Mart customer can pay by credit
card (A), debit card (B), cash (C) or check (D).
Credit
Card
DebitCard
Cash
Check
Col lect ively Exhaus t ive Events
Rules of Probability
R l f P b bili
-
7/28/2019 Doane Chapter 05A
40/58
Polytomous events can be made dichotomous
(binary) by defining the second category as
everything notin the first category.
Polytomous Events Binary(Dichotomous) Variable
Vehicle type (SUV, sedan, truck,
motorcycle)
X= 1 if SUV, 0 otherwise
Forced Dicho tomy
Rules of Probability
A randomly-chosen NBA players
height
X= 1 if height exceeds 7 feet, 0
otherwiseTax return type (single, married filing
jointly, married filing separately, head
of household, qualifying widower)
X= 1 if single, 0 otherwise
R l f P b bili
-
7/28/2019 Doane Chapter 05A
41/58
The probability of eventAgiven that event B has
occurred.
Denoted P(A | B).The vertical line | is read as given.
( )
( | ) ( )
P A BP A B
P B
forP(B) > 0 and
undefined otherwise
Condit ional Probabi l i ty
Rules of Probability
R l f P b bilit
-
7/28/2019 Doane Chapter 05A
42/58
Consider the logic of this formula by looking at the
Venn diagram.( )
( | )( )
P A BP A B
P B
The sample space is
restricted to B, an event
that has occurred.
A B is the part ofBthat is also inA.
The ratio of the relative
size ofA B to B isP(A | B).
Condit ional Probabi l i ty
Rules of Probability
R l f P b bilit
-
7/28/2019 Doane Chapter 05A
43/58
Of the population aged 16 21 and not in college:
Unemployed 13.5%
High school dropouts 29.05%
Unemployed high school dropouts 5.32%
What is the conditional probability that a memberof this population is unemployed, given that the
person is a high school dropout?
Example: High School Dropouts
Rules of Probability
R l f P b bilit
-
7/28/2019 Doane Chapter 05A
44/58
First define
U= the event that the person is unemployed
D = the event that the person is a high school
dropout
P(U) = .1350 P(D) = .2905 P(UD) = .0532
( ) .0532( | ) .1831
( ) .2905
P U DP U D
P D
or 18.31%
P(U | D) = .1831 > P(U) = .1350
Therefore, being a high school dropout is related
to being unemployed.
Example: High School Dropouts
Rules of Probability
I d d t E t
-
7/28/2019 Doane Chapter 05A
45/58
EventA is independentof event B if the conditionalprobability P(A | B) is the same as the marginal
probability P(A).
To check for independence, apply this test:
IfP(A | B) = P(A) then eventA is independentofB.
Another way to check for independence:
IfP(A B) = P(A)P(B) then eventA isindependentof event B since
P(A | B) = P(A B) = P(A)P(B) = P(A)P(B) P(B)
Independent Events
I d d t E t
-
7/28/2019 Doane Chapter 05A
46/58
Out of a target audience of 2,000,000, adA
reaches 500,000 viewers, B reaches 300,000
viewers and both ads reach 100,000 viewers.
What is P(A | B)?
500,000( ) .25
2,000,000P A
300,000( ) .15
2,000,000P B
100,000( ) .05
2,000,000P A B
Independent Events
Example: Televis ion Ads
( ) .05( | ) .30
( ) .15
P A BP A B
P B
.3333 or 33%
Independent Events
-
7/28/2019 Doane Chapter 05A
47/58
So, P(adA) = .25
P(ad B) = .15
P(AB) = .05P(A | B) = .3333
P(A | B) = .3333 P(A) = .25
P(A)P(B)=(.25)(.15)=.0375 P(AB)=.05
Are eventsA and B independent?
Independent Events
Example: Televis ion Ads
Independent Events
-
7/28/2019 Doane Chapter 05A
48/58
When P(A) P(A | B), then eventsA and B are
dependent.
For dependent events, knowing that event B has
occurred will affect theprobabilitythat eventA willoccur.
For example, knowing a persons age would affecttheprobabilitythat the individual uses text
messaging but causation would have to be proven
in other ways.
Independent Events
Dependent Events
Statistical dependence does not prove causality.
Independent Events
-
7/28/2019 Doane Chapter 05A
49/58
An actuarystudies conditional probabilities
empirically, using
- accident statistics
- mortality tables
- insurance claims records
Many businesses rely on actuarial services, so a
business student needs to understand the
concepts of conditional probability and statistical
independence.
Independent Events
Actuar ies Again
Independent Events
-
7/28/2019 Doane Chapter 05A
50/58
The probability ofn independent events occurring
simultaneously is:
To illustrate system reliability, suppose a Web site
has 2 independent file servers. Each server has
99% reliability. What is the total system reliability?Let,
P(A1A2...An) = P(A1) P(A2) ... P(An)
if the events are independent
F1 be the event that server 1 fails
F2be the event that server 2 fails
Independent Events
Mult ip l ication Law for Independent Events
Independent Events
-
7/28/2019 Doane Chapter 05A
51/58
Applying the rule of independence:
The probability that at least one server is up is:
P(F1F2) = P(F1) P(F2)= (.01)(.01) = .0001
1 - .0001 = .9999 or 99.99%
So, the probability that both servers are down is
.0001.
Independent Events
Mult ip l ication Law for Independent Events
Independent Events
-
7/28/2019 Doane Chapter 05A
52/58
Redundancycan increase system reliability even
when individual component reliability is low.
NASA space shuttle has three independent flightcomputers (triple redundancy).
Each has an unacceptable .03 chance of failure
(3 failures in 100 missions).
Let Fj= event that computerjfails.
Independent Events
Example: Space Shu tt le
Independent Events
-
7/28/2019 Doane Chapter 05A
53/58
What is the probability that all three flight
computers will fail?
P(all 3 fail) = P(F1F2F3)
= 0.000027
or 27 in 1,000,000 missions.
= P(F1) P(F2) P(F3) presuming
that failures
are
independent
= (0.03)(0.03)(0.03)
Independent Events
Example: Space Shu tt le
Independent Events
-
7/28/2019 Doane Chapter 05A
54/58
How high must reliability be?
Public carrier-class telecommunications data links
are expected to be available 99.999% of the time.
The five nines rule implies only 5 minutes of
downtime per year.
This type of reliability is needed in many businesssituations.
Independent Events
The Five Nines Ru le
Independent Events
-
7/28/2019 Doane Chapter 05A
55/58
For example,
Independent Events
The Five Nines Ru le
Independent Events
-
7/28/2019 Doane Chapter 05A
56/58
Suppose a certain network Web server is up only 94% of the
time. What is the probability of it being down?
How many independent servers are needed to ensure that
the system is up at least 99.99% of the time (or down only
1 - .9999 = .0001 or .01% of the time)?
P(down) = 1P(up) = 1 .94 = .06
Independent Events
How Much Redundancy is Needed?
Independent Events
-
7/28/2019 Doane Chapter 05A
57/58
So, to achieve a 99.99% up time, 4 redundant
servers will be needed.
2 servers: P(F1F2) = (0.06)(0.06) = 0.0036
3 servers: P(F1F2F3)
= (0.06)(0.06)(0.06) = 0.0002164 servers: P(F1F2F3F4)
= (0.06)(0.06)(0.06)(0.06)
=0.00001296
Independent Events
How Much Redundancy is Needed?
-
7/28/2019 Doane Chapter 05A
58/58
Applied Statistics inBusiness and Economics
End of Part 1 of Chapter 5
top related