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Chapter 2. Foundations of Probability

Section 2.2. Sample Space and Events

Jiaping Wang

Department of Mathematical Science

01/14/2013, Monday

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Outline

Syllabus

Randomness Sample Space and Events

Event Operator and Venn Diagram

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Part 1. Syllabus

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Syllabus: Part I

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Syllabus: Part II

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Syllabus: Part III

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Part 2. Randomness

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Randomness around us

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Part 3. Sample Space and Events

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Flip a coin: {Head, Tail}

Sample Space

Mutually Exclusive means the outcomes of the set do not overlap.Exhaustive means the list contains all possible outcomes.

Definition 2.1A sample space S is a set that includes all possible outcomes for a random experimentlisted in a mutually exclusive and exhaustive way.

Roll a die: {1, 2, 3, 4, 5, 6}

President Election: {Tom, Jerry}

Measuring your height: {X|X>0}

Final Grades: {X|0<=X<=100}

Letter Grades: {A, B, C, D, F}

Roll two dice: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (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)

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For example, if we have a sample space S={1,2,3}, then the possible events are Φ, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, S where we can say Φ is the impossible Event or null event and S is called sure event.

Definition 2.2:An event is any subset of a sample space.

Events

In an experiment, we are interested in a particular outcome or a subset of outcomes.

If a sample space has n elements, then there are 2n possible subsets.

For a die-rolling experiment, S={1,2,3,4,5,6},A is the event of “ an even number”: A={2,4,6}B is the event of “an odd number”: B={1,3,5}C is the event of “greater than or equal to 5”: C={ 5, 6}D is the event of “less than 1”: D= Φ

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Part 4. Event Relations and Venn Diagram

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The white area in S is the complement of AUBUC, denoted as AUBUC

For sample space S, there are events A, B, and C, then the new eventsD=A∩C, E=B∩C, F=A ∩B, G=A∩B∩CHere G is a subset of either D, E, or F.

Event Operators and Venn Diagram

There are three operators between events:Intersection: ∩ --- A∩B or AB – a new event consisting of common

elements from A and BUnion: U --- AUB – a new event consisting of all outcomes from A or B.Complement: ¯, A, -- a subset of all outcomes in S that are not in A.

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AUB

Event Operators and Venn Diagram (Cont.)

A∩B A

S SS

Additional Operators

Relative Complement B\A=A∩ B

Symmetric Difference A∆B=(A\B) U (B\A)

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Some Laws

Commutative laws:

Associate laws:

Distributive laws:

DeMorgan’s laws:

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An Example

If we have a sample space S={1, 2, 3, 4, 5, 6}, there are two events A={1, 2, 5}, B={2, 3, 4}, then

A U B = {1, 2, 3, 4, 5} A ∩ B = {2}

B = {1, 5, 6}

B\A = { 3, 4}

A ∩ B = {6}

A = {3, 4, 6}

A \B = {1, 5}

A ∆ B = {1, 3, 4, 5}

A U B =S\(A UB)= {6}

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Example 2.2

After a hurricane, 50% of the residents in a particular Florida county were without electricity, 47% were without water and 38% were without telephone service. Although20% of the residents had all three utilities, 10% were without all three, 12% were withoutelectricity and water but still had a working telephone, and 4% were without electricity andworking telephone but still had water. A county resident is randomly selected and whetheror not the person was without water, electricity or telephone service after the hurricane is recorded. Express each of the following events in set notation and find the percentage of county residents represented by each.

Q1: The selected county resident was without electricity only but had water and telephone service.

Q2: The selected county resident was without only one of the utilities but had the other two.Q3: The selected county resident was without exactly two of the utilities.

Q4: The selected county resident was without at least one of the utilities.

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Solutions

Sample Space: S={all residents in the particular Florida county}Determine the events based on the utilities:

A={residents with electricity}, B={residents with water}, C={residents with phone service}

Find useful information: A has 50%, B has 53% and C has 62%, A∩B∩C has 20%, AUBUC has 10%, A ∩ B ∩ C has 12%, A ∩ C ∩B has 4%.

Q1: (B∩C)\A= B∩C∩A = A – AUBUC - A ∩ B ∩ C - A ∩ C ∩B has 24%

Q2: (A∩B∩ C) U (A∩B∩ C) U (A∩B∩ C) has 24%+6%+5%=35%

Q3: (A∩B∩ C) U (A∩B∩ C) U (A∩B∩ C) has 19%+12%+4%=35%

Q4: AUBUC = A∩B∩C has 1-20%=80%