Chapter 6 ReviewChapter 6 Review

Fr Chris ThielFr Chris Thiel

13 Dec 200413 Dec 2004

What is true about probability?

• The probability of any event must be a number between 0 and 1 inclusive

• The sum of all the probabilities of all outcomes in the sample space must be 1

• The probability of an event is the sum of the outcomes in the sample space which make up the event

Independent

Previous outcomes do not change probability

Multiplication Rule: P(A and B)=P(A)P(B)

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Iff P(B | A) = P(B)

Disjoint

One outcome precludes the other since there isNo overlap…

Complement

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Ac = A = notA = −A

The event A does not occur

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P(Ac ) =1− P(A)

Addition Rules

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

Multiplication Rules

P(A and B)=P(A)P(B) if A and B are independent

Conditional Rules

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P(A∩ B) = P(A)P(B | A)

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P(B | A) =P(A∩ B)

P(A)

P(65+)=18%P(Widowed)=10%

a. If among 65+, 44% widowed, What percent of the population are widows over 65?

b. If 8% are widows over 65, What is the chance of being a widow given that they’re over 65?

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P(65 + ∩W ) = P(65+)P(W | 65+)

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=(.18)(.44) = .0792

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P(W | 65+) =P(W ∩ 65+)

P(65+)

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=.08

.18≈ .44

See Table 6.1 p. 366

Use Venn Diagrams & Trees

Venn Diagrams can help see if events are Independent, complementary or disjoint

Use Tree Diagrams to Organize addition andMultiplication rules to combinations of events

If event A and B are disjoint, then

• P(A and B)= 0

• P(A or B) =1

• P(B)=1-P(A)

Independent events… you flip a coin and it’s heads 4 times in a row…. The odds

are STILL the same

The 6 is 3 times more likely to occur… what is

the probability of rolling a 1 or a 6?

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x + x + x + x + x + 3x =1

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38 + 1

8 = 12

A fair die is tossedA fair die is tossed4 or 5-win $14 or 5-win $1

6-win $46-win $4 If you play twice:If you play twice:

what is the probability that you will win what is the probability that you will win $8?$8?

$2?$2?

P(A)=.5P(B)=.6

P(A andB)=.1

• P(A|B)=?

• Are A and B Independent?

• Disjoint?

• Will either A or B always occur?

• Are A and B complementary?

A B

.4 .1 .5

0

Lie Detector• Reports “Lie” 10% if person is telling

the truth

• Reports “Lie” 95% if the person is actually lying

• Probability of machine never reporting a lie if 5 truth tellers use it

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(.9)5 = .59049

You enter a lottery, the odds of getting a prize

is .11If you try 5 times, what is

the probability that you will win at least once?

• 1-P(never winning)

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1− (.89)5

.08

.92

.96

.04

.07

.93

Has Disease

No Disease

Tests +

Tests -

Tests +

Tests -

(.08)(.96)=.0768

(.92)(.07)=.0644

(.08)(.04)=.0032

(.92)(.93)=.85

8% have a disease. A test detects the disease 96%And falsely indicates the disease 7%. If you test positive, what is the chance you have the disease?

P(D|+)

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P(D | +) =P(D∩ +)

P(+)

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=(.0768)

(.0768) + (.0644)≈ .65

P(Harvard)=40%P(Florida)=50%P(both)=20%P(none)=?P(F but not H)=?

H F

.2.2 .3

.3

30% of calls result in a airline reservation.a. P(10 calls w/o a reservation)=?

b. P(at least 1 out of 10 calls has a reservation)=?

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(1− .3)10 ≈ .0282

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=1− P(none) =1− .0282 = .9718

85% fire calls are for medical emergenciesAssuming independence…

P(exactly one of two calls is for a medical emergency)=?

P(M)P(F)+P(F)P(M)=(.85)(.15)+(.15)(.85)=.255

Is it really independent?