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### Chapter3

Probability

Objectives

- Develop probability as a measure of uncertainty
- Introduce basic rules for finding probabilities
- Use probability as a measure of reliability for an inference

Events, Sample Spaces and Probability

- Experiment – process of observation that leads to a single outcome with no predictive certainty
- Sample point – most basic outcome of an experiment
- Sample Space – a listing of all sample points for an experiment

Events, Sample Spaces and Probability: Examples

Experiment 1: tossing a die

- What are sample points?
- What is the sample space?
Experiment 2: tossing 2 coins

- What are sample points?
- What is the sample space?

HH

TH

HT

TT

S

Sample Points

Name of Sample Space

Events, Sample Spaces and Probability- Venn Diagram

Events, Sample Spaces and Probability

Probability Rules for Sample Points:

- All sample point probabilities must lie between 0 and 1
- The sum of all sample point probabilities must be one

Events, Sample Spaces and Probability

How to assign Sample Point Probabilities?

- Relative Frequency Method: Assigning probabilities based on experimentation or historical data.
- Classical/Theoretical Method: Assigning probabilities based on the assumption of equally likely outcomes.
- Subjective Method: Assigning probabilities based on the assignor’s judgment.

Events, Sample Spaces and Probability

- An Event is a specific collection of sample points
- The Probability of an event is the sum of the probabilities of all sample points in the sample space for the event

Events, Sample Spaces and Probability

- How to assign Event Probabilities?
- Define experiment
- List sample points
- Assign probabilities to sample points
- Identify collection of sample points in Event
- Sum sample point probabilities

1,2

1,3

1,4

1,5

1,6

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

1/36

2,1

2,2

2,3

2,4

2,5

2,6

3,1

3,4

3,2

3,3

3,5

3,6

4,1

4,2

4,3

4,4

4,5

4,6

5,1

5,2

5,3

5,4

5,5

5,6

6,1

6,2

6,3

6,4

6,5

6,6

Events, Sample Spaces and Probability: Example- What is the probability of rolling an eight in a single toss of a pair of dice?
- Experiment is toss of pair of dice

- Probability of rolling an 8 = 1/36+1/36+1/36+1/36+1/36 = 5/36= .14

Events, Sample Spaces and Probability: Example (continued)

- What is the probability of rolling at least a 9 with a single toss of two dice?
P(at least 9) = P(9) + P(10) + P(11) + P(12)

= 4/36 + 3/36 + 2/36 + 1/36

= 10/36

= 5/18

= .28

Events, Sample Spaces and Probability

- What do you do when the number of sample points is too large to enumerate?
- Use the Combinations Rule to count the number of samples possible when selecting a sample of size n from N elements
- where

Events, Sample Spaces and Probability: Example

- In the dice throwing example, there are 36 pairings. How many different samples of 2 pairs can we select from those 36 pairs?
- We have 630 possible samples of 2 pairs from a group of 36 pairs

Events, Sample Spaces and Probability

- If you had 30 people interested in being in a study and you needed 5, how many different combinations of 5 are there?

Unions and Intersections

- A Compound Event is a composition of 2 or more events
- Can be the result of a union or intersection of events

Unions and Intersections

- The union of events A and B is the event containing all sample points that are in A or B or both.
- The intersection of two events A and B is the event consisting of all sample points that are in both A and B.

Unions and Intersections:Example

- Event A: A New Jersey Birth Mother is white
- Event B: A New Jersey Birth Mother was a teenager when giving birth

Complementary Events

- The complement of an Event A is the event AC consisting of all sample points that do not belong to Event A

Complementary Events:Example

- If A is having at least 1 head appear in the toss of 2 coins, AC is having no heads appear

The Additive Rule

- The probability that event A or B will occur when an experiment is performed is given by
P( )=P(A) + P(B) – P( )

- Example:

Mutually Exclusive Events

- Events are mutually exclusive if they share no sample points.

The Additive Rule and Mutually Exclusive Events

- The Additive Rule for Mutually Exclusive Events

Applying the Concepts: Examples

- Problem 3.44, p. 142
- Problem 3.49, p. 143
- Problem 3.48, p. 143

Conditional Probability

- Conditional Probability is the probability that event A occurs given that event B occurs
- Conditional probability works with a reduced sample space, the space that contains B
and

Conditional Probability

- Event A: cause of complaint is appearance
- Event B: complaint occurred during guarantee period

Conditional Probability

- Problem 3.68, p. 156

Conditional Probability:Example

- Study of Ancient Pottery:
Of the 837 pottery pieces uncovered at the excavation site, 183 were painted. These painted pieces included 14 painted in a curvilinear decoration, 165 painted in a geometric decoration, and 4 painted in a naturalistic decoration. Suppose one of the 837 pottery pieces is selected and examined.

a. What is the probability that the pottery piece is painted?

b. Given that the pottery piece is painted, what is the probability that it is painted in a curvilinear decoration?

The Multiplicative Rule

- The probability that both A and B will occur when an experiment is performed is given by

The Multiplicative Rule and Independent Events

- Events A and B are independent if the occurrence of one does not alter the probability of the other occurring
- Test for Independent Events:
A and B are independent events if either

or

The Multiplicative Rule and Independent Events: Example 1

- Event A: cause of complaint is appearance
- Event B: complaint occurred during guarantee period
- Are A and B independent events?

The Multiplicative Rule and Independent Events: Example 2

- Problem 3.80, p. 158
- Refer to problem 3.83, p. 158

Random Sampling

- Assume a desired sample size of n
- Sample is random if every set of n elements in the population has the same probability of being selected
- Random number generators often used to produce a random sample

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