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Introduction to ProbabilityProbability Arithmetic and Conditional Probability

Chapter 4BA 201

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PROBABILITY ARITHMETIC

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The addition law provides a way to compute the probability of event A, or B, or both A and B occurring.

Addition Law

The law is written as:

P(A B) = P(A) + P(B) - P(A B

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Event M = Markley Oil Profitable

Event C = Collins Mining Profitable

M C = Markley Oil Profitable or Collins Mining Profitable

We know: P(M) = 0.70, P(C) = 0.48, P(M C) = 0.36

Thus: P(M C) = P(M) + P(C) - P(M C)

= 0.70 + 0.48 - 0.36

= 0.82

Addition Law

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

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Scenario

Outcome Probability

O1 0.10

O2 0.30

O3 0.05

O4 0.15

O5 0.20

O6 0.05

O7 0.10

O8 0.05

Event

Outcomes Probability

E1 O1, O3 0.15

E2 O1, O4, O5, O6

0.50

E3 O2 0.30

E4 O7, O8 0.15

E5 O4, O5, O7 0.45

E6 O1, O7 0.20

A statistical experiment has the following outcomes, along with their probabilities and the following events, with the corresponding outcomes.

Outcomes Events

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Addition Law

Using the addition law, what is the probability of E2 ⋃ E5 [P(E2 ⋃ E5)]?

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The probability of an event given that another event has occurred is called a conditional probability.

A conditional probability is computed as follows :

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

Conditional Probability

( )( | )

( )P A B

P A BP B

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Conditional Probability

If you draw a card from a deck of cards, what is the probability of a Jack given you draw a face card, P(Jack|Face Card)?There are 12 face cards, 4 of which are Jacks.

P(Jack|Face Card) = 4/12 = 1/3

Count and Divide

P(Jack ⋂ Face Card) = 4/52 (since all Jacks are face cards)P(Face Card) = 12/52

P(Jack|Face Card) = P(Jack ⋂ Face Card) / P(Face Card)

Computed

= (4/52)/(12/52) = 1/3

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Event M = Markley Oil Profitable

Event C = Collins Mining Profitable

We know: P(M C) = 0.36, P(M) = 0.70

Conditional Probability

Thus: ( ) .36

( | ) .5143( ) .70

P C MP C M

P M

= Collins Mining Profitable given Markley Oil Profitable

( | )P C M

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Conditional Probability

What is the probability of E5 given E4 [P(E5| E4)]? P(E5 ⋂ E4)=0.10, P(E4)=0.15

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Conditional Probability

What is the probability of E5 given E4 [P(E5| E4)]? P(E5 ⋂ E4)=0.10, P(E4)=0.15

E4E5

O4, O5 O7 O8

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Multiplication Law

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

The law is written as:

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

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

OR

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Event M = Markley Oil ProfitableEvent C = Collins Mining Profitable

We know: P(M) = 0.70, P(C|M) = 0.5143

Multiplication Law

M C = Markley Oil Profitable and Collins Mining Profitable

Thus: P(M C) = P(M)P(M|C)= (0.70)(0.5143)

= 0.36(This result is the same as that obtained earlierusing the definition of the probability of an event.)

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Multiplication Law

Using the multiplication rule, what is the probability of E2

⋂ E5 [P(E2 ⋂ E5)]? P(E2|E5)=0.7778, P(E5)=0.45 (OR) P(E5|E2)=0.70, P(E2)=0.50

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Joint Probability Table

Collins MiningProfitable (C) Not Profitable (Cc)Markley Oil

Profitable (M)

Not Profitable (Mc)

Total 0.48 0.52

Total

0.70

0.30

1.00

0.36 0.34

0.12 0.18

Joint Probabilities(appear in the

bodyof the table)

Marginal Probabilities(appear in the

marginsof the table)

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Joint Probability Table

Collins MiningProfitable (C) Not Profitable (Cc)Markley Oil

Profitable (M)

Not Profitable (Mc)

Total 0.48 0.52

Total

0.70

0.30

1.00

0.36a 0.34c

0.12b 0.18d

P(M ⋂ C) = 0.20+0.16=0.36a

P(Mc ⋂ C) = 0.10+0.02=0.12b

P(M ⋂ Cc) = 0.08+0.26=0.34c

P(Mc ⋂ Cc) = 0.12+0.06=0.18d

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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

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Independent Events

A bag contains three marbles, 1 blue and 2 red. If you draw a red marble, what is the probability the next marble you draw is blue?

Blue = 1/3

Red= 2/3

Blue = 0

Red= 1

Blue = 1/2

Red= 1/2

P(Blue)=1/3

P(Blue|Red)=1/2

P(Red)=2/3

P(Red|Blue)=1

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Independent Events

If you flip a coin and get a head, what is the probability of getting a tail on the next flip?

Head= 1/2

Tail=1/2

Head=1/2

Tail=1/2

Head=1/2

Tail=1/2

P(Head)=1/2

P(Head|Tail)=1/2

P(Tail)=1/2

P(Tail|Head)=1/2

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The multiplication law also can be used as a test to see if two events are independent.

The law is written as:

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

Multiplication Lawfor Independent Events

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Event M = Markley Oil Profitable

Event C = Collins Mining Profitable

We know: P(M C) = 0.36, P(M) = 0.70, P(C) = 0.48

But: P(M)P(C) = (0.70)(0.48) = 0.34, not 0.36

Are events M and C independent?

DoesP(M C) = P(M)P(C) ?

Hence: M and C are not independent.

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Multiplication Lawfor Independent Events

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Multiplication Law for Independent Events

Eh1=Head on first flip. Eh2=Head on second flip. P(Eh1 ⋂ Eh2)

Head= 1/2

Tail=1/2

Head=1/2

Tail=1/2

Head=1/2

Tail=1/2

H, H=1/4=P(Eh1 ⋂ Eh2)

P(Eh1)=1/2

P(Eh2)=1/2

P(Eh1) P(Eh2)=1/4

P(Eh1 ⋂ Eh2)=1/4

P(Eh1 ⋂ Eh2)=P(Eh1) P(Eh2)

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Do not confuse the notion of mutually exclusive events with that of independent events.

Two events with nonzero probabilities cannot be both mutually exclusive and independent.

If one mutually exclusive event is known to occur, the other cannot occur; thus, the probability of the other event occurring is reduced to zero (and they are therefore dependent).

Mutual Exclusiveness and Independence

Two events that are not mutually exclusive, might or might not be independent.

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