Precalculus with Limits: A Graphing Approach

7th Edition

ISBN: 9781305071711

Author: Ron Larson

Publisher: Brooks/Cole

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Question

Chapter 8.6, Problem 75E

(a)

To determine

Whether $A$ and $B$ are mutually exclusive, and to draw a diagram to support answer.

(a)

Expert Solution

Answer to Problem 75E

Thus for mutually exclusive event is

$1.34$

Explanation of Solution

Given information:

Let $A$ and $B$ be two events from the same sample space such that $P(A)=0.76$ and $P(B)=0.58$ .

Calculation:

Consider two events $A$ and $B$ from the same sample space such that

$P(A)=0.76$ and $P(B)=0.58$

Event $A$ and $B$ are exclusive if $A$ and $B$ have no sample point in common or if

$A∩B$ is empty.

If $A∩B$ is empty, then

$P(A∩B)=P(ϕ)=0$

Thus for mutually exclusive event is

$P(A∪B)=P(A)+P(B)−P(A∩B)=0.76+0.58−0=1.34$

But the probability $P(A∪B)$ is not possible since the probability of any event cannot be greater than one

Thus, the events $A$ and $B$ are not mutually exclusive since the sum of probability is greater than one

The diagram showing the events $A$ and $B$ is shown below.

Precalculus with Limits: A Graphing Approach, Chapter 8.6, Problem 75E , additional homework tip 1

(b)

To determine

Whether $A′$ and $B′$ are mutually exclusive, and to draw a diagram to support answer.

(b)

Expert Solution

Answer to Problem 75E

The sum of probability of the complementary events $A′$ and $B′$ is

$0.66$

Explanation of Solution

Given information:

Let $A$ and $B$ be two events from the same sample space such that $P(A)=0.76$ and $P(B)=0.58$ .

Calculation:

The formula of the complement of the event is

$P(X′)=1−P(X)$

Therefore, the complement of $P(A)$ and $P(B)$ is

$P(A′)=1−P(A)=1−0.76=0.24$

And

$P(B′)=1−P(B)=1−0.58=0.42$

Thus, the sum of probability of the complementary events $A′$ and $B′$ is

$P(A′)$ and $P(B′)$

$=0.24+0.42=0.66$

Thus, the events $A′$ and $B′$ are mutually exclusive since the sum of probability is less than one

The diagram showing the events $A′$ and $B′$ is shown below

Precalculus with Limits: A Graphing Approach, Chapter 8.6, Problem 75E , additional homework tip 2

(c)

To determine

To find: the possible range of $P(A∪B)$ .

(c)

Expert Solution

Answer to Problem 75E

The range of $P(A∪B)$ is $0.76≤P(A∪B)≤1$

Explanation of Solution

Given information:

Let $A$ and $B$ be two events from the same sample space such that $P(A)=0.76$ and $P(B)=0.58$ .

Calculation:

Note that the probability of an event $A$ or $B$ is always greater than the individual probabilities of the events $A$ and $B$

That is

$P(A∪B)≥P(A) and P(A∪B)≥P(B)$

Thus,

$P(A∪B)≥0.76$

Since the probability of an event is always smaller than 1, therefore

$P(A∪B)≤1$

Therefore, the range of $P(A∪B)$ is $0.76≤P(A∪B)≤1$

Chapter 8.6, Problem 74E

Chapter 8.6, Problem 76E

Chapter 8 Solutions

Precalculus with Limits: A Graphing Approach

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Prob. 67E Ch. 8.6 - Prob. 68E Ch. 8.6 - Prob. 69E Ch. 8.6 - Prob. 70E Ch. 8.6 - Prob. 71E Ch. 8.6 - Prob. 72E Ch. 8.6 - Prob. 73E Ch. 8.6 - Prob. 74E Ch. 8.6 - Prob. 75E Ch. 8.6 - Prob. 76E Ch. 8.6 - Prob. 77E Ch. 8.6 - Prob. 78E Ch. 8.6 - Prob. 79E Ch. 8.6 - Prob. 80E Ch. 8 - Prob. 1RE Ch. 8 - Prob. 2RE Ch. 8 - Prob. 3RE Ch. 8 - Prob. 4RE Ch. 8 - Prob. 5RE Ch. 8 - Prob. 6RE Ch. 8 - Prob. 7RE Ch. 8 - Prob. 8RE Ch. 8 - Prob. 9RE Ch. 8 - Prob. 10RE Ch. 8 - Prob. 11RE Ch. 8 - Prob. 12RE Ch. 8 - Prob. 13RE Ch. 8 - Prob. 14RE Ch. 8 - Prob. 15RE Ch. 8 - Prob. 16RE Ch. 8 - Prob. 17RE Ch. 8 - Prob. 18RE Ch. 8 - Prob. 19RE Ch. 8 - Prob. 20RE Ch. 8 - Prob. 21RE Ch. 8 - Prob. 22RE Ch. 8 - Prob. 23RE Ch. 8 - Prob. 24RE Ch. 8 - Prob. 25RE Ch. 8 - Prob. 26RE Ch. 8 - Prob. 27RE Ch. 8 - Prob. 28RE Ch. 8 - Prob. 29RE Ch. 8 - Prob. 30RE Ch. 8 - Prob. 31RE Ch. 8 - Prob. 32RE Ch. 8 - Prob. 33RE Ch. 8 - Prob. 34RE Ch. 8 - Prob. 35RE Ch. 8 - Prob. 36RE Ch. 8 - Prob. 37RE Ch. 8 - Prob. 38RE Ch. 8 - Prob. 39RE Ch. 8 - Prob. 40RE Ch. 8 - Prob. 41RE Ch. 8 - Prob. 42RE Ch. 8 - Prob. 43RE Ch. 8 - Prob. 44RE Ch. 8 - Prob. 45RE Ch. 8 - Prob. 46RE Ch. 8 - Prob. 47RE Ch. 8 - Prob. 48RE Ch. 8 - Prob. 49RE Ch. 8 - Prob. 50RE Ch. 8 - Prob. 51RE Ch. 8 - Prob. 52RE Ch. 8 - Prob. 53RE Ch. 8 - Prob. 54RE Ch. 8 - Prob. 55RE Ch. 8 - Prob. 56RE Ch. 8 - Prob. 57RE Ch. 8 - Prob. 58RE Ch. 8 - Prob. 59RE Ch. 8 - Prob. 60RE Ch. 8 - Prob. 61RE Ch. 8 - Prob. 62RE Ch. 8 - Prob. 63RE Ch. 8 - Prob. 64RE Ch. 8 - Prob. 65RE Ch. 8 - Prob. 66RE Ch. 8 - Prob. 67RE Ch. 8 - Prob. 68RE Ch. 8 - Prob. 69RE Ch. 8 - Prob. 70RE Ch. 8 - Prob. 71RE Ch. 8 - Prob. 72RE Ch. 8 - Prob. 73RE Ch. 8 - Prob. 74RE Ch. 8 - Prob. 75RE Ch. 8 - Prob. 76RE Ch. 8 - Prob. 77RE Ch. 8 - Prob. 78RE Ch. 8 - Prob. 79RE Ch. 8 - Prob. 80RE Ch. 8 - Prob. 81RE Ch. 8 - Prob. 82RE Ch. 8 - Prob. 83RE Ch. 8 - Prob. 84RE Ch. 8 - Prob. 85RE Ch. 8 - Prob. 86RE Ch. 8 - Prob. 87RE Ch. 8 - Prob. 88RE Ch. 8 - Prob. 89RE Ch. 8 - Prob. 90RE Ch. 8 - Prob. 91RE Ch. 8 - Prob. 92RE Ch. 8 - Prob. 93RE Ch. 8 - Prob. 94RE Ch. 8 - Prob. 95RE Ch. 8 - Prob. 96RE Ch. 8 - Prob. 97RE Ch. 8 - Prob. 98RE Ch. 8 - Prob. 99RE Ch. 8 - Prob. 1CT Ch. 8 - Prob. 2CT Ch. 8 - Prob. 3CT Ch. 8 - Prob. 4CT Ch. 8 - Prob. 5CT Ch. 8 - Prob. 6CT Ch. 8 - Prob. 7CT Ch. 8 - Prob. 8CT Ch. 8 - Prob. 9CT Ch. 8 - Prob. 10CT Ch. 8 - Prob. 11CT Ch. 8 - Prob. 12CT Ch. 8 - Prob. 13CT Ch. 8 - Prob. 14CT Ch. 8 - Prob. 15CT Ch. 8 - Prob. 16CT Ch. 8 - Prob. 17CT Ch. 8 - Prob. 18CT Ch. 8 - Prob. 19CT Ch. 8 - Prob. 20CT Ch. 8 - Prob. 21CT Ch. 8 - Prob. 22CT Ch. 8 - Prob. 23CT Ch. 8 - Prob. 24CT Ch. 8 - Prob. 25CT Ch. 8 - Prob. 26CT

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Whether A and B are mutually exclusive, and to draw a diagram to support answer.

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Whether A<m:math><m:mi>A</m:mi></m:math> and B<m:math><m:mi>B</m:mi></m:math> are mutually exclusive, and to draw a diagram to support answer.

Answer to Problem 75E

Explanation of Solution

Whether A′<m:math><m:msup><m:mi>A</m:mi><m:mo>′</m:mo></m:msup></m:math> and B′<m:math><m:msup><m:mi>B</m:mi><m:mo>′</m:mo></m:msup></m:math> are mutually exclusive, and to draw a diagram to support answer.

Answer to Problem 75E

Explanation of Solution

To find: the possible range of P(A∪B)<m:math><m:mrow><m:mi>P</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>A</m:mi><m:mo>∪</m:mo><m:mi>B</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow></m:math> .

Answer to Problem 75E

Explanation of Solution

Chapter 8 Solutions

Whether $A$ and $B$ are mutually exclusive, and to draw a diagram to support answer.

Whether $A′$ and $B′$ are mutually exclusive, and to draw a diagram to support answer.

To find: the possible range of $P(A∪B)$ .