Module 1 Probability

8/13/2019 Module 1 Probability

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Module 1

Probability


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Review of Sets

Definitions Union of Sets (AUB): Elements in either set

Intersection (AB): Elements in both sets

Complement (Ac): Elements not in A

Rules A(BUC) = (AB)U(AC)

AU(BC) = (AUB)(AUC)

n(A) = n(AB) + n(ABc)

n(AUB) = n(A) + n(B)n(AB)

n(AUBUC) = n(A) + n(B) + n(C)n(AB)n(AC)n(BC) + n(ABC)


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Solving Problems

Method

Draw diagram of intersecting sets

Label known quantities Work from the inside out to determine unknowns

Example 1

N = 100, A = 40, B = 30, AB = 20

Find ABc= 4020 = 20

Find (AUB)c= 100(40 + 3020) = 50


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Solving Problems

Example 2 A = 40, B = 30, C = 20

AB = 20, AC = 10, BC = 10

ABC = 5

Calculations ABCc= ABABC = 205 = 15

A

Bc

C = 105 = 5 Ac

B

C = 105 = 5 ABcCc= AABCcABcCABC = 15

AcBCc= 5 AcBcC = 5

AUBUC = (40 + 30 + 20)(20 + 10 + 10) + 5 = 55


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Using Table

Use table if Acis as important as A For instance, not just male, but male and female

Or looking for values such as (AcBc)

Method Draw table with rows representing one variable and

columns representing another.

Systematically calculate desired values using set rules:

n(A) = Nn(Ac)

n(ABc) = n(A)n(AB)


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Example Problem

Males: 500 (Given)

Male & College: 350 (Given) Male & No College

Male & Imm: 100 (Given) 80 (Given) 100 - 80 = 20

M & Non-I: 500 - 100 = 400 350 - 80 = 270 400 - 279 = 130

Imm & College: Not Needed

A survey of 1000 students: 500 students are male 600 plan to attend college

200 are immigrants 350 are males/college

150 are imm/college 100 are immigrant males 80 are male/immigrants/college

How many students are male, non-immigrants, who donot plan to attend college?


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Probability Definitions

Sample Point: Simple outcome of experiment

Rolling a 4 on a die

Sample Space (S): All possible outcomes Set for rolling a die: {1, 2, 3, 4, 5, 6}

Mutually Exclusive: Never at same time (AB)=0

Can not role a 3 and an even number at same time

Exhaustive: Include entire sample space

Even and odd rolls include all possible outcomes


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Probability Definitions (Same as Sets)

Event: Collection of sample points

Event A: Rolling an even number with die: {2, 4, 6}

Union (AUB): Outcomes in events A or B A: {2, 4, 6}, B: {4, 5, 6} AUB = {2, 4, 5, 6}

Intersection (AB): Outcomes in events A and B

A: {2, 4, 6}, B: {4, 5, 6} AB = {4, 6}

Complement (Ac): Outcomes not in A

A: {2, 4, 6} Ac: {1, 3, 5}


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Probability Definition

Probability RulesP(Ac) = 1P(A)

Law of Total Probability

Probability Definition and Rules

Note: Aiare mutually exclusive and exhaustive


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Other useful Rules

DeMorgans Laws (AUB)c= AcBc

(AB)c= AcUBc

Example: Not (even or > 3) = odd and (


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Example

A local survey indicated

80% own auto 60% own house 50% own both

What percentage own an auto or a house,but not both?

What percentage own neither an auto nor a house?

Solution

P(ABc)+P(A

cB) = (.80.50) + (.60.50) = .40

P(AcBc) = P[(AUB)c] = 1(.8 + .6.5) = .1


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Conditional Probability

Definitions

Example

Law of Total Probability


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Sample Problem

A survey of company clients indicated 30% have life insurance 70% medical 20% both

50% of those with only life will renew policy

40% of those with only medical will renew

60% with both will renew

Determine probability that a random policy holder will renew?

Method Draw Venn diagram and calculate regions.

Set up equation for conditional probability.

Use conditional definition to change and probability to conditional.


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Solution

Note: Looking for conditional probability that person will renew(E), given that they hold a policy (AUBUC). A, B, and C aremutually exclusive, so equations simplify.

A = LMc P[A] = P[L]P[LM] = .3 - .2 = .1

B = LcM P[B] = P[M]P[LM] = .7 - .2 = .5

C = LM P[C] = P[LM] = .2


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Bayes Theorem

Useful when swapping conditional probability

Example

Two urns have 5 red and 6 blue balls. A ball fromurn 1 is picked randomly and placed in urn 2.

If a red ball is pulled from urn 2, what is theprobability that a red ball was picked from urn 1?


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Solution

Calculate Probabilities

P[R1] = 5/11 P[B1] = 6/11

P[R2|R1] = 6/12 P[R2|B1] = 5/12

Use Bayes Theorem


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Sample Problem

Given the following data, what is the probability of a personhaving an accident falling into the 2630 bracket?

Age of Driver Probability ofAccident

Portion ofDrivers

Group 1: 18-20 .07 .03Group 2: 21-25 .05 .05

Group 3: 26-30 .03 .10

Group 4: 31-65 .02 .52

Group 5: >65 .05 .30


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Independence

Events are independent if the probability of one isnot affected by the other.

Independence Equations P[AB] = P[A] P[B]

P[A|B] = P[A]

Example

P[A] = P[B] = A&B independent

P[AUB] = P[A] + P[B]P[AB] = P[A]+ P[B]P[A]*P[B]

P[AUB] = + - = 5/8

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Module 1 Probability