Chapter 4

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McGraw-Hill/Irwin

Chapter 4

Probability

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Chapter Outline

4.1 The Concept of Probability4.2 Sample Spaces and Events4.3 Some Elementary Probability

Rules4.4 Conditional Probability and

Independence4.5 Bayes’ Theorem (Optional)

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4.1 The Concept of Probability

An experiment is any process of observation with an uncertain outcome

The possible outcomes for an experiment are called the experimental outcomes

Probability is a measure of the chance that an experimental outcome will occur when an experiment is carried out

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Probability

If E is an experimental outcome, then P(E) denotes the probability that E will occur and:Conditions

1. 0 P(E) 1 such that: If E can never occur, then P(E) = 0 If E is certain to occur, then P(E) = 1

2. The probabilities of all the experimental outcomes must sum to 1

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Assigning Probabilities to Experimental Outcomes

Classical MethodFor equally likely outcomes

Long-run relative frequency In the long run

SubjectiveAssessment based on experience,

expertise or intuition

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4.2 Sample Spaces and Events

Sample Space: The set of all possible experimental outcomes

Sample Space Outcomes: The experimental outcomes in the sample space

Event: A set of sample space outcomes

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Events

If A is an event, then 0 ≤ P(A) ≤ 1

1. If an event never occurs, then the probability of this event equals 0

2. If an event is certain to occur, then the provability of this event equals 1

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Example 4.3

Figure 4.2

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4.3 Some Elementary Probability Rules

1. Complement2. Union3. Intersection4. Addition5. Conditional probability

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Complement

Figure 4.4

APAP 1

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Union and Intersection

1. The intersection of A and B are elementary events that belong to both A and B

Written as A ∩ B

2. The union of A and B are elementary events that belong to either A or B or both

Written as A B

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Union and Intersection Diagram

Figure 4.5

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Mutually Exclusive

Figure 4.6

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The Addition Rule

If A and B are mutually exclusive, then the probability that A or B will occur is

P(AB) = P(A) + P(B)

If A and B are not mutually exclusive:

P(AB) = P(A) + P(B) – P(A∩B)

where P(A∩B) is the joint probability of A and B both occurring together

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4.4 Conditional Probability and Independence

The probability of an event A, given that the event B has occurred, is called the conditional probability of A given BDenoted as P(A|B)

Further, P(A|B) = P(A∩B) / P(B)P(B) ≠ 0

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Interpretation

Restrict sample space to just event B

The conditional probability P(A|B) is the chance of event A occurring in this new sample space

In other words, if B occurred, then what is the chance of A occurring

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General Multiplication Rule

BAPBP

ABPAPBAP

|

|

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Independence of Events

Two events A and B are said to be independent if and only if:

P(A|B) = P(A)

This is equivalently to

P(B|A) = P(B)

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The Multiplication Rule

The joint probability that A and B (the intersection of A and B) will occur is

P(A∩B) = P(A) P(B|A) = P(B) P(A|B)

If A and B are independent, then the probability that A and B will occur is:

P(A∩B) = P(A) P(B) = P(B) P(A)

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4.5 Bayes’ Theorem (Optional)

S1, S2, …, Sk represents k mutually exclusive possible states of nature, one of which must be true

P(S1), P(S2), …, P(Sk) represents the prior probabilities of the k possible states of nature

If E is a particular outcome of an experiment designed to determine which is the true state of nature, then the posterior (or revised) probability of a state Si, given the experimental outcome E, is calculated using the formula on the next slide

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Example 4.18

Oil drilling on a particular siteP(S1 = none) = .7P(S2 = some) = .2P(S3 = much) = .1

Can perform a seismic experimentP(high|none) = .04P(high|some) = .02P(high|much) = .96

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Example 4.18 Continued

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Chapter 4

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