probability addition rule

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MARIO F. TRIOLA

Essentials of STATISTICS Section 3-3 Addition Rule

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Example: Let event A represent a woman and B represent blue eyes.

A or B is a compound event representing the set of people who are women or who have blue eyes.

A and B is a compound event that represents the set of people who are women AND have blue eyes (i.e. blue eyed women)

A compound event is any event that combines two or more simple events.

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Venn Diagrams

Women blue eyes

Let the circle on the left represent the characteristic “subject is a woman”

Let the circle on the right represent the characteristic “subject has blue eyes”

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Venn Diagrams– “or”

Women blue eyes

Together, the blue and yellow circles representthe compound event “is a woman or has blue eyes.”

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Venn Diagrams

Women blue eyes

The intersection of the two circles represents the event “subject is a woman and has blue eyes.”

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A={accountant} B = {is a downhill skier}

A or B ?

A and B?

Interpret these Events:

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A={accountant} B = {is a downhill skier}

A or B ? Is an accountant or is a downhill skier

A and B?

Is an accountant and a downhill skier

Interpret these Events:

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When finding the probability that event A occurs or event B occurs, find

1. the total number of ways A can occur and

2. the number of ways B can occur,

3. but find the total in such a way that no outcome is counted more than once.

General Addition Rule

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P(A or B) = P(A) + P(B) - P(A and B)

where P(A and B) denotes the probability that A and B both occur at the same time.

Formal Addition Rule

If P(A) = .5 and P(B) = .2 and P(A and B) = .1,

then,

P(A or B) = 0.5 + 0.2 – 0.1 = 0.6

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Example: Addition Rule

In a study of 82 young drivers, 39 were men who at some point in time were ticketed for traffic violations. 11 were men who were never ticketed. 8 were women who were ticketed, and 24 were women who were never ticketed.

If one of these subjects is randomly selected, what is the probability of:

(a)Getting a man or someone who was ticketed?

(b)Getting a man or someone who was NOT ticketed?

(c)Getting a woman or someone who was ticketed?

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Example: Addition Rule

In a study of 82 young drivers, 39 were men who at some point in time were ticketed for traffic violations. 11 were men who were never ticketed. 8 were women who were ticketed, and 24 were women who were never ticketed.

P(man) = 50 men / 82 surveyed = 0.6098

P(woman) = 32 women / 82 surveyed = 0.3902

P(ticketed) = 47 ticketed / 82 surveyed = 0.5732

P(never ticketed) = 35 never ticketed / 82 surveyed = 0.4268

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Example: Addition Rule

If one of these subjects is randomly selected, what is the probability of:

(a) Getting a man or someone who was ticketed?P(man or ticketed) = P(man) + P(ticketed) – P(man and ticketed)

= 50/82 + 47/82 - 39/82 = 58/82

(b) Getting a man or someone who was NOT ticketed?P(man or not ticketed) = P(man) + P(not ticketed) –

P(man and not ticketed)

= 50/82 + 35/82 – 11/82 = 74/82

(c) Getting a woman or someone who was ticketed? P(woman or ticketed) = P(woman) + P(ticketed) –

P(woman and ticketed)

= 32/82 + 47/82 - 8/82 = 71/82

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Mutually ExclusiveEvents A and B are mutually exclusive if they

cannot occur simultaneously.

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Mutually ExclusiveEvents A and B are mutually exclusive if they

cannot occur simultaneously.

P(A) P(B)

P(A and B)

NOT mutually exclusive!

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DefinitionEvents A and B are mutually exclusive if they

cannot occur simultaneously.

P(A) P(B)

Events A and B

are

Mutually exclusive

Nothing in common

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Mutually Exclusive Events

If two events A and B are mutually exclusive, they cannot occur at the same time.

The implication is that it is impossible for these two events A and B to occur together, so

P(A and B) = 0

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Mutually Exclusive Events

If two events are mutually exclusive, this simplifies the addition rule.

P(A or B) = P(A) + P(B) - P(A and B)

= P(A) + P(B) - 0

= P(A) + P(B)

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Applying the Addition Rule

P(A or B)

Addition Rule

AreA and Bmutuallyexclusive

?

P(A or B) = P(A)+ P(B) - P(A and B)

P(A or B) = P(A) + P(B)Yes

No

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Find the probability of randomly selecting a man or a boy.

Men Women Boys Girls Totals

Survived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 56 2223

Contingency Table

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Find the probability of randomly selecting a man or a boy.

Men Women Boys Girls Totals

Survived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 56 2223

Contingency Table

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Find the probability of randomly selecting a man or a boy.

P(man or boy) = 1692 + 64 = 1756 = 0.7902223 2223 2223

Men Women Boys Girls Totals

Survived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 56 2223

Contingency Table

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Find the probability of randomly selecting a man or a boy.

P(man or boy) = 1692 + 64 = 1756 = 0.7902223 2223 2223

Men Women Boys Girls Totals

Survived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 56 2223

Contingency Table

* Mutually Exclusive *

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Find the probability of randomly selecting a man or someone who survived.

Men Women Boys Girls Totals

Survived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 56 2223

Contingency Table

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Find the probability of randomly selecting a man or someone who survived.

Men Women Boys Girls Totals

Survived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 56 2223

Contingency Table

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Find the probability of randomly selecting a man or someone who survived.

P(man or survivor) = 1692 + 706 - 332 = 1756 2223 2223 2223 2223

Men Women Boys Girls Totals

Survived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 56 2223

Contingency Table

= 0.929

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Find the probability of randomly selecting a man or someone who survived.

P(man or survivor) = 1692 + 706 - 332 = 1756 2223 2223 2223 2223

Men Women Boys Girls Totals

Survived 332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 56 2223

Contingency Table

* NOT Mutually Exclusive *

= 0.929

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Complementary Events

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Complementary Events

P(A) and P(A)are

mutually exclusive

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Complementary Events

P(A) and P(A)are

mutually exclusive

All simple events are either in A or A.

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Complementary EventsIf two events are mutually exclusive, then they cannot occur at the same time.

The probabilities of these two events will make 1.

P(A) + P(A) = 1

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Rules of Complementary Events

P(A) + P(A) = 1

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P(A)

Rules of Complementary Events

P(A) + P(A) = 1

= 1 - P(A)

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P(A) + P(A) = 1

= 1 - P(A)

P(A) = 1 - P(A)

P(A)

Rules of Complementary Events

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Example:

When a baby is born in the United States, the true probability that the child is a boyis 0.512. What is the true probability that the child is a girl?

The events are complementary. Onecannot be a boy and a girl at the same time.

P(girl) = 1 – P(boy) = 1 – 0.512 = 0.488

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Example:

A poll shows that 61% of Americans saythey believe that life exists elsewhere in the galaxy. What is the probability that someone does not have that belief?

The events are complementary. One cannot believe in life and not believe in life at the same time.

P(not) = 1 – P(believes) = 1 – 0.61 = 0.39

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Example:

Your alarm clock has a 97.5% probabilityof working on any given morning. What is the probability that it will NOT work?

The events are complementary. The alarmcannot “work” and “not work” at the sametime.

P(not) = 1 – P(works) = 1 – 0.975 = 0.025

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Example:Suppose your use the principle of redundancyto wake you up in the morning. You setTWO alarms, both with 97.5% probabilityof working on any given morning. What is the probability that your system works?

The events “works” and “fails” are two complementary events.

P(works) = 1 – P(both fail) = 1 – (0.025)(0.025) = 0.999375

The redundant alarm system is almost certainly going to work every morning.