EC2:Quantitative Methods for Business Decisions

CONDITIONAL PROBABILITY

Example:Of 200 people interviewed regarding intention to buy your prod-uct and whether are not they are financially able to do so, thefollowing results were obtained:

To Buy (B) Not to Buy (B)

Able to Finance (A) 40 20 60Unable to Finance (A) 80 60 140

120 80 200

Which may be expressed as

To Buy (B) Not to Buy (B)

Able to Finance (A) .2 .1 .3Unable to Finance (A) .4 .3 .7

.6 .4

What is the probability that

1. a customer is financially able and has a desire to buy?(P (A ∩B))

2. a customer has a desire to buy? (P (B))

3. a customer will buy given that she has the financial abilityto pay? (conditional probability P (B|A) )

Conditional Probability

DEFN: The conditional probability of B given A is

P (B|A) =P (A ∩B)

P (A)

Terminology:

• Joint Probability: P (A ∩B)

• Marginal Probability: P (A), P(B)

• Conditional Probability: P (A|B) or P (B|A)

A rearrangement of the above definition yields the following:

Multiplication Law of Probability:

Two events

P (A)P (B|A)P (A ∩B) =

P (B)P (A|B)

More than two events:

P (E1 ∩ E2 ∩ E3 · · · ∩ Ek) =

P (E1)P (E2|E1)P (E3|E1 ∩ E2) · · ·P (Ek|E1 ∩ E2 · · · ∩ Ek)

Examples:

1. Consider four firms A, B, C, D bidding for a certain con-tract. A survey of past bidding success of these firms showthe following probabilities of winning:

P (A) = 0.35, P (B) = 0.15, P (C) = 0.3, P (D) = 0.2

Before the decision is made to award the contract, firm Bwithdraws the bid. Find the new probabilities of winningthe bid for A, C and D.

2. Pull three cards from a deck without replacement. What isthe probability that all are black?

3. A firm submits tenders for two different contracts. Theprobability that the first and second tenders will be success-ful is 50% and 30% respectively. Calculate the probabilitythat:

(a) both will be successful;

(b) neither will be successful;

(c) only the first will be successful;

(d) at least one will be successful.

Independent Events

DEFN: Independent Events

A and B are said to be independent if

P (A|B) = P (A)

Multiplication Law

Two Independent Events

P (A ∩B) = P (A)P (B)

More than Two Independent Events

P (E1 ∩ E2 ∩ E3 · · ·Ek) = P (E1)P (E2) · · ·P (Ek)

Law of Total Probability

Examples:

1. In a certain company

50% of documents are written in WORD;30% in WORDSTAR;20% in WORDPERFECT.

From past experience it is know that:

40% of the WORD documents exceed 10 pages20% of the WORDSTAR exceed 10 pages20% of the WORDPERFECT documents exceed 10 pages

(a) What is the overall proportion of documents containingmore than 10 pages?

2. An insurance company runs three different offices, A, B andC. 30% of the company’s employees work in Office A, 20%work in Office B and 50% work in Office C. 10% of the staffin Office A are managers, 20% of the staff in Office B aremanagers and 5% from Office C are managers.

Offices A B C

Proportion Employees .3 .2 .5Proportion Managers .1 .2 .5

(a) What is the total proportion of managers in the com-pany?

Law of Total ProbabilityIf a sample space can be partitioned into k mutually exclu-sive and exhaustive events:

A1, A2, A3, · · ·Ak

i.e.S = A1 ∪ A2 ∪ A3 · · · ∪ Ak

Then for any event E:

P (E) = P (A1)P (E|A1)+P (A2)P (E|A2) · · ·P (Ak)P (E|AK)

Proof:

E = E ∩ S

= E ∩ (A1 ∪ A2 ∪ · · · ∪ Ak)

= (E ∩ A1) ∪ (E ∩ A2) ∪ · · · (E ∩ Ak)

Since these are mutually exclusive

P (E) = P (E ∩ A1) + P (E ∩ A2) + · · ·P (E ∩ Ak)

= P (A1)P (E|A1) + P (A2)P (E|A2) + · · ·P (Ak)P (E|Ak)

Example 1

1. In a certain company

50% of documents are written in WORD;30% in WORDSTAR;20% in WORDPERFECT.

From past experience it is know that:

40% of the WORD documents exceed 10 pages20% of the WORDSTAR exceed 10 pages20% of the WORDPERFECT documents exceed 10 pages

(a) What is the overall proportion of documents containingmore than 10 pages?

.5 ∗ .4 + .3 ∗ .2 + .2 ∗ .2 = .3

Tree Diagram

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