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Example: Bradley InvestmentsExample: Bradley Investments

Bradley has invested in two stocks, Markley Oil and Bradley has invested in two stocks, Markley Oil and Collins Mining. Bradley has determined that theCollins Mining. Bradley has determined that thepossible outcomes of these investments three possible outcomes of these investments three

monthsmonthsfrom now are as follows.from now are as follows.

Investment Gain or LossInvestment Gain or Loss in 3 Months (in $000)in 3 Months (in $000)

Markley OilMarkley Oil Collins MiningCollins Mining

1010 55 002020

8822

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Bradley Investments can be viewed as aBradley Investments can be viewed as atwo-step experiment. It involves two stocks, two-step experiment. It involves two stocks, eacheachwith a set of experimental outcomes.with a set of experimental outcomes.

Markley Oil:Markley Oil: nn11 = 4 = 4

Collins Mining:Collins Mining: nn22 = 2 = 2Total Number of Total Number of

Experimental Outcomes:Experimental Outcomes: nn11nn22 = (4)(2) = 8 = (4)(2) = 8

A Counting Rule for A Counting Rule for Multiple-Step ExperimentsMultiple-Step Experiments

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Tree DiagramTree Diagram

Gain 5Gain 5

Gain 8Gain 8

Gain 8Gain 8

Gain 10Gain 10

Gain 8Gain 8

Gain 8Gain 8

Lose 20Lose 20

Lose 2Lose 2

Lose 2Lose 2

Lose 2Lose 2

Lose 2Lose 2

EvenEven

Markley OilMarkley Oil(Stage 1)(Stage 1)

Collins MiningCollins Mining(Stage 2)(Stage 2)

ExperimentalExperimentalOutcomesOutcomes

(10, 8) (10, 8) Gain $18,000 Gain $18,000

(10, -2) (10, -2) Gain $8,000 Gain $8,000

(5, 8) (5, 8) Gain $13,000 Gain $13,000

(5, -2) (5, -2) Gain $3,000 Gain $3,000

(0, 8) (0, 8) Gain $8,000 Gain $8,000

(0, -2) (0, -2) Lose Lose $2,000$2,000

(-20, 8) (-20, 8) Lose Lose $12,000$12,000

(-20, -2)(-20, -2) Lose Lose $22,000$22,000

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Subjective MethodSubjective Method

Applying the subjective method, an analyst Applying the subjective method, an analyst made the following probability assignments.made the following probability assignments.

Exper. OutcomeExper. OutcomeNet Gain Net Gain oror Loss Loss ProbabilityProbability(10, 8)(10, 8)(10, (10, 2)2)(5, 8)(5, 8)(5, (5, 2)2)(0, 8)(0, 8)(0, (0, 2)2)((20, 8)20, 8)((20, 20, 2)2)

$18,000 Gain$18,000 Gain $8,000 Gain$8,000 Gain $13,000 Gain$13,000 Gain $3,000 Gain$3,000 Gain $8,000 Gain$8,000 Gain $2,000 Loss$2,000 Loss $12,000 Loss$12,000 Loss $22,000 Loss$22,000 Loss

.20.20

.08.08

.16.16

.26.26

.10.10

.12.12

.02.02

.06.06

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Events and Their ProbabilitiesEvents and Their Probabilities

Event Event MM = Markley Oil Profitable = Markley Oil Profitable

MM = {(10, 8), (10, = {(10, 8), (10, 2), (5, 8), (5, 2), (5, 8), (5, 2)}2)}

PP((MM) = ) = PP(10, 8) + (10, 8) + PP(10, (10, 2) + 2) + PP(5, 8) + (5, 8) + PP(5, (5, 2)2)

= .20 + .08 + .16 + .26= .20 + .08 + .16 + .26

= .70= .70

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Events and Their ProbabilitiesEvents and Their Probabilities

Event Event CC = Collins Mining Profitable = Collins Mining Profitable

CC = {(10, 8), (5, 8), (0, 8), ( = {(10, 8), (5, 8), (0, 8), (20, 8)}20, 8)}

PP((CC) = ) = PP(10, 8) + (10, 8) + PP(5, 8) + (5, 8) + PP(0, 8) + (0, 8) + PP((20, 8)20, 8)

= .20 + .16 + .10 + .02= .20 + .16 + .10 + .02

= .48= .48

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Union of Two EventsUnion of Two Events

Event Event MM = Markley Oil Profitable = Markley Oil Profitable

Event Event CC = Collins Mining Profitable = Collins Mining Profitable

MM CC = Markley Oil Profitable = Markley Oil Profitable oror Collins Mining Profitable Collins Mining Profitable

MM CC = {(10, 8), (10, = {(10, 8), (10, 2), (5, 8), (5, 2), (5, 8), (5, 2), (0, 8), (2), (0, 8), (20, 8)}20, 8)}

PP((MM C)C) = = PP(10, 8) + (10, 8) + PP(10, (10, 2) + 2) + PP(5, 8) + (5, 8) + PP(5, (5, 2)2)

+ + PP(0, 8) + (0, 8) + PP((20, 8)20, 8)

= .20 + .08 + .16 + .26 + .10 + .02= .20 + .08 + .16 + .26 + .10 + .02

= .82= .82

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Intersection of Two EventsIntersection of Two Events

Event Event MM = Markley Oil Profitable = Markley Oil Profitable

Event Event CC = Collins Mining Profitable = Collins Mining Profitable

MM CC = Markley Oil Profitable = Markley Oil Profitable andand Collins Mining Profitable Collins Mining Profitable

MM CC = {(10, 8), (5, 8)} = {(10, 8), (5, 8)}

PP((MM C)C) = = PP(10, 8) + (10, 8) + PP(5, 8)(5, 8)

= .20 + .16= .20 + .16

= .36= .36

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Event Event MM = Markley Oil Profitable = Markley Oil ProfitableEvent Event CC = Collins Mining Profitable = Collins Mining Profitable

MM CC = Markley Oil Profitable = Markley Oil Profitable oror Collins Mining Profitable Collins Mining Profitable

We know: We know: PP((MM) = .70, ) = .70, PP((CC) = .48, ) = .48, PP((MM CC) = .36) = .36

Thus: Thus: PP((MM C) C) = = PP((MM) + P() + P(CC) ) PP((MM CC))

= .70 + .48 = .70 + .48 .36 .36

= .82= .82

Addition LawAddition Law

(This result is the same as that obtained earlier(This result is the same as that obtained earlierusing the definition of the probability of an event.)using the definition of the probability of an event.)

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Event Event MM = Markley Oil Profitable = Markley Oil Profitable

Event Event CC = Collins Mining Profitable = Collins Mining Profitable

We know:We know: P P((MM CC) = .36, ) = .36, PP((MM) = .70 ) = .70

Thus: Thus:

Conditional ProbabilityConditional Probability

( ) .36( | ) .5143

( ) .70P C M

P C MP M

( ) .36( | ) .5143

( ) .70P C M

P C MP M

= Collins Mining Profitable= Collins Mining Profitable givengiven Markley Oil Profitable Markley Oil Profitable

( | )P C M( | )P C M

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Event Event MM = Markley Oil Profitable = Markley Oil ProfitableEvent Event CC = Collins Mining Profitable = Collins Mining Profitable

We know:We know: P P((MM) = .70, ) = .70, PP((CC||MM) = .5143) = .5143

Multiplication LawMultiplication Law

MM CC = Markley Oil Profitable = Markley Oil Profitable andand Collins Mining Profitable Collins Mining Profitable

Thus: Thus: PP((MM C) C) = = PP((MM))PP((M|CM|C))= (.70)(.5143)= (.70)(.5143)

= .36= .36

(This result is the same as that obtained earlier(This result is the same as that obtained earlierusing the definition of the probability of an event.)using the definition of the probability of an event.)

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Multiplication LawMultiplication Lawfor Independent Eventsfor Independent Events

Event Event MM = Markley Oil Profitable = Markley Oil ProfitableEvent Event CC = Collins Mining Profitable = Collins Mining Profitable

We know:We know: P P((MM CC) = .36, ) = .36, PP((MM) = .70, ) = .70, PP((CC) = .48) = .48 But: But: PP((M)P(C) M)P(C) = (.70)(.48) = .34, not .36= (.70)(.48) = .34, not .36

Are events Are events MM and and CC independent? independent?DoesDoesPP((MM CC) = ) = PP((M)P(C) M)P(C) ??

Hence:Hence: M M and and CC are are notnot independent. independent.