Bradley has invested in two stocks, Markley Oil

Bradley has invested in two stocks, Markley Oil

and Collins Mining. Bradley has determined that the

possible outcomes of these investments three months

from now are as follows. Investment Gain or Loss in 3 Months (in $000)

Markley Oil Collins Mining 10 5 0-20

8-2

Example: Bradley Investments

The number of experiment outcomes

+5

+8

+8

+10

+8

+8

-20

-2

-2

-2

-2

0

Markley Oil(Stage 1)

Collins Mining(Stage 2)

ExperimentalOutcomes


(10 + 8)1000 = $18,000

(10 – 2)1000 = $8,000

(5 + 8)1000 = $13,000

(5 – 2)1000 = $3,000

(0 + 8)1000 = $8,000

(0 – 2)1000 = –$2,000

(-20 + 8)1000 = –$12,000

(-20 – 2)1000 = –$22,000

8


(50)(50)(50)(50)(501 15,625,)(50) 000 01!

,00

Example: State lotteries

Politicians propose a new lottery. In this lottery there are 6 jars each filled with 50 ping pong balls numbered 1 to 50. How many distinct winning tickets could win this lottery if you have to pick the order in which they come out of the jars?

Since order matters and there is replacement the total number of experimental outcomes equals


1 11,441,304(50)(49) ,000(48)(47)( 15,890,7006 5 4 3 2 1

46

)(!

6 45)


Politicians propose a new lottery. In this lottery there is one jar filled with 50 ping pong balls numbered 1 to 50. How many distinct winning tickets could win this lottery if the order of the balls does not have to be picked?

Since order does not matter and there is no replacement the total number of experimental

outcomes equals

!!

( )!Nn

NCNn n

-



Since order matters and there is no replacement the total number of experimental outcomes equals

(50)(49)(48)(47)(461 11,441,)(45) 304 01!

,00

Politicians propose a new lottery. In this lottery there is one jar filled with 50 ping pong balls numbered 1 to 50. Six balls are selected one at a time. How many distinct winning tickets could win this lottery if the order of the balls must be chosen too?

1!!

( )!N

nNP

N n

-


Probability

0 1.5

Increasing Likelihood of Occurrence

Probability:

P(A U B) = P(A) + P(B) if A & B are ME

P(A ∩ B) = P(A) P(B) if A & B are Indep.

If an experiment has n possible outcomes, the

classical method would assign a probability of 1/n

to each outcome.Experiment: Rolling a die

Sample Space: S = {1, 2, 3, 4, 5, 6}

Probabilities: Each sample point has a 1/6 chance of occurring

Example: Rolling a Die

Probability

(50)(50)(50)(50)(501 15,625,)(50) 000 01!

,00

Example: State lottery 1

Since order matters and there is replacement the total number of experimental outcomes equals

1( ) 0.00000000006415,625,000,000

P win

Politicians propose a new lottery. In this lottery there are 6 jars each filled with 50 ping pong balls numbered 1 to 50. What is the probability of winning this lottery if you have to pick the order in which they come out of the jars?

Probability

1 11,441,304(50)(49) ,000(48)(47)( 15,890,7006 5 4 3 2 1

46

)(!

6 45)

Since order does not matter and there is no replacement the total number of experimental

outcomes equals

1( ) 0.000000062915,890,700

P win

Example: State lottery 2Politicians propose a new lottery. In this lottery there is

one jar filled with 50 ping pong balls numbered 1 to 50. What is the probability of winning this lottery if the order of the balls does not have to be picked?

Probability

Since order matters and there is no replacement the total number of experimental outcomes equals

1( ) 0.000000000087411,441,304,000

P win

(50)(49)(48)(47)(461 11,441,)(45) 304 01!

,00

Example: State lottery 3Politicians propose a new lottery. In this lottery there is

one jar filled with 50 ping pong balls numbered 1 to 50. Six balls are selected one at a time. What is the probability of winning this lottery if the order of the balls must be chosen too?

Probability

Example: US population by age (The World Almanac 2004) is given below:

AGE Frequency RelativeLL UL (millions) Frequency0 19 80.5 0.29

20 24 19 0.0725 34 39.9 0.1435 44 45.2 0.1645 54 37.7 0.1355 64 24.3 0.0965 35 0.12Totals 281.6 1.00

Probability

Probability






8-2


Markley discovers 3 oil reserves under the ocean using its 3 R&D

vessels.

.10

Probability






8-2



vessels.

.10

.25

Probability






8-2



vessels.

.10

.25

.40

Probability






8-2



vessels.

.10

.25

.40

.25

Probability






8-2


.10

.25

.40

.25

.80

The FED keeps interest rates set a

0.25%

Probability






8-2


The FED raises interest rates to

2.50%

.10

.25

.40

.25

.80

.20

Probability

+5

+8

+8

+10

+8

+8

-20

-2

-2

-2

-2

0

Markley Oil(Stage 1)

Collins Mining(Stage 2)

ExperimentalOutcomes


$18,000

$8,000

$13,000

$3,000

$8,000

–$2,000

–$12,000

–$22,000

Probability(.10)(.80) =.08(.10)(.20) =.02(.25)(.80) =.20(.25)(.20) =.05(.40)(.80) =.32(.40)(.20) =.08(.25)(.80) =.20(.25)(.20) =.05

1.00

Probability

Example: US population by ageLet A be the event “55 years of age or older.” Compute the probability of Ac.

Complement of an Event

( ) 0.09 0.12 0.21P A

( ) 0.29 0.07 0.14 0.16 0.13 0.79cP A

( ) 1 ( ) 1 0.21 0.79cP A P A - -

AGE RelativeLL UL Frequency0 19 0.29

20 24 0.0725 34 0.1435 44 0.1645 54 0.1355 64 0.0965 0.12

Totals 1.00

( ) 0.09P A B

Example: US population by ageLet A be the event: “55 years of age or older.” Let B be the event:“64 years of age or younger.” Compute the probability of A and B.

( ) 0.29 0.07 0.14 0.16 0.13 0.880.09P B

0.09( ) 0.12 0.21P A

Intersection of Two Events


20 24 0.0725 34 0.1435 44 0.1645 54 0.1355 64 0.0965 0.12

Totals 1.00

( ) 0CP A

Example: US population by age

( ) 0.29 0.07 0.36P C

( ) 0.09 0.12 0.21P A

Intersection of Mutually Exclusive Events

Let A be the event: “55 years of age or older.” Let C be the event:“24 years of age or younger.” Compute the probability of A and C.

0.36


20 24 0.0725 34 0.1435 44 0.1645 54 0.1355 64 0.0965 0.12

Totals 1.00

( ) ( ) ( )0.21 0.8 98

( )0.0

1

P AP A B P A P BB - -


( ) 0.88P B

( ) ( ) 0.21 0.88 1.09P A P B

Union of Two Events

( ) 0.21P A

Let A be the event: “55 years of age or older.” Let B be the event:“64 years of age or younger.” Compute the probability of A or B.


20 24 0.0725 34 0.1435 44 0.1645 54 0.1355 64 0.0965 0.12

Totals 1.00

( ) ( ) ( )0.21 0.360

0.

( )

57

P AP A C P A P C C - -


Union of Mutually Exclusive Events

Let A be the event “55 years of age or older.” Let C be the event“24 years of age or younger.” Compute the probability of A or C.

( ) 0.29 0.07 0.36P C

( ) 0.09 0.12 0.21P A


20 24 0.0725 34 0.1435 44 0.1645 54 0.1355 64 0.0965 0.12

Totals 1.00

The conditional probability of A given B is denoted by P(A|B).

Conditional Probability

( )( | ) ( )P A BP A B

P B

700( ) 0.7143980

P W

Conditional ProbabilityExample: Consider the hiring of black and white workers at

BigMart

White (W)

Black (B) Total

Hired (H) 130 30 160

Not Hired (Hc) 570 250 820

Total 700 280 980

( ) 0.1327( | ) 0.1858( ) 0.7143

P H WP H WP W

130( ) 0.1327980

P H W

130( | ) 0.1858700

P H W

700( ) 0.7143980

P W

Conditional ProbabilityExample: Consider the hiring of black and white workers at

BigMart

White (W)

Black (B) Total

Hired (H) 130 30 160

Not Hired (Hc) 570 250 820

Total 700 280 980

130( ) 0.1327980

P H W

( )( | ) ( )P A BP A B

P B

( )( ) ( | ) ( )( )

P A BP B P A B P BP B

( ) ( | ) ( )P B P A B P A B ( ) ( ) ( | )P A B P B P A B

Multiplication Law For Dependent Events

The multiplication law provides a way to compute the probability of the intersection of two events.

It is derived by manipulating the conditional probability:

White (W)

Black (B) Total

Hired (H) 130 30 160

Not Hired (Hc) 570 250 820

Total 700 280 980

Example: Consider the hiring of black and white workers at BigMart

700 130( ) ( ) ( | ) 0.1327980 700

P H W P W P H W

( ) 0.1327P H W

Multiplication Law For Dependent Events

Independent Events

If the probability of event A is not changed by the existence of event B, we would say that events A and B are independent.

Two events A and B are independent if:

P(A|B) = P(A) P(B|A) = P(B)or

( ) ( ) ( )P A B P B P A

This changes the multiplication law:

( ) ( ) ( | )P A B P B P A B

Example: Consider the promotion status of economists at some economic research think tank:

Men (M)

Women (W) Total

Promoted (P) 100 20 120

Not Promoted (N) 500 100 600

Total 600 120 720

100( | ) 0.1667600

P P M 120( ) 0.1667720

P P

Independent Events

“Getting the promotion” and “being male” are independent events

160( ) 0.1633980

P H

Example: Consider the hiring of black and white workers at BigMart

White (W)

Black (B) Total

Hired (H) 130 30 160

Not Hired (Hc) 570 250 820

Total 700 280 980

30( | ) 0.1071280

P H B

Independent Events

“Getting hired” and “being black” are not independent eventsdata_simpson.xls

http://www.halsnarr.com/ppt/data_simpson.xlsx

Bayes’ Theorem

Tells us about how the probability of something changeswhen we learn information.

For example, we know from a drug lab’s claim:

P(testing positive | employee is a druggie)

Since we fire druggies, we want to know:

P(employee is a druggie | testing positive)

To compute the latter we need additional information (e.g.,the false positive rate, prevalence of drug use among our employees)

We want to ensure that our employees are not taking drugs because this is a safety risk.

We contract with a laboratory that claims their drug test is 94% accurate but there is a 5% chance of a false positive.

Suppose we have 10,000 employees and that 1% of them are druggies. If an employee is found to be a druggie, we fire them for safety reasons.

P = test is PositiveN = test is NegativeD = employee is a DruggieDc = employee is NOT a Druggie

Bayes’ Theorem

Example: Cocaine drug testing

From the drug lab we know:P(P | D) = 0.94


Bayes’ Theorem

(accuracy of the test)

P(N | D) = 0.06

P(P | Dc) = 0.05 (false positive rate)

P(N | Dc) = 0.95

We believe:P(D) = 0.01 (prevalence)

P(Dc) = 0.99

P(N|D) = .06P(D) = .01

P(Dc) = .99P(P|Dc) = .05

P(N|Dc) = .95

P(P|D) = .94 P(P D) = .0094

P(P Dc) = .0495

P(N Dc) = .9405

P(N D) = .0006

Prevalence Drug Lab ExperimentalOutcomes

P(P) = .0589

P(N) = .9411

Bayes’ Theorem


Bayes’ Theorem

( )( )

( | )( | ) ( ) (| ) | )( c cP P D

P PP D P

P D PDP PD

DP

D

To find the (posterior) probability that an employee is a druggiegiven he tested positive, we apply Bayes’ theorem.

The drug lab’s accuracy claim

The proportion of our employees that are not

druggies.

The drug lab’s false positive rate.

Prevalence of drug use in our

company.


Q1 What is the probability a worker is a druggie given he tested positive for cocaine use?

(.01)(.94)(.01)(.94) (.99)(.05)

= .1596

Bayes’ Theorem

( ) ( | )( | ) ( ) ( | ) ( ) ( | )c cP D P P DP D P

P D P P D P D P P D


.0094.0094 .0495

.0094

.0589

Q2 What is the probability a worker isn’t a druggie given he tested positive for cocaine use?

Q3 What is the probability a worker is a druggie given he did not test positive for cocaine use?

Q4 What is the probability a worker is NOT a druggie given he did not test positive for cocaine use?

Bayes’ Theorem


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Bradley has invested in two stocks, Markley Oil

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