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Some parts of California are particularly earthquake-prone. Suppose that in one metropolitan area. 25% of all homeowners are insured against earthquake damage. Four homeowners arc to be selected at random: let X denote the number among the four who have earthquake insurance. a. Find the probability distribution of X . [ Hint: Let S denote a homeowner who has insurance and F one who does not. Then one possible outcome is SFSS, with probability (.25)(.75)(.25)(.25) and associated X value 3. There are 15 other outcomes.] b. Draw the corresponding probability histogram. c. What is the most likely value for X ? d. What is the probability that at least two of the four selected have earthquake insurance?

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Probability and Statistics for Eng...

9th Edition
Jay L. Devore
Publisher: Cengage Learning
ISBN: 9781305251809

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Chapter
Section
BuyFindarrow_forward

Probability and Statistics for Eng...

9th Edition
Jay L. Devore
Publisher: Cengage Learning
ISBN: 9781305251809
Chapter 3.2, Problem 16E
Textbook Problem
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Some parts of California are particularly earthquake-prone. Suppose that in one metropolitan area. 25% of all homeowners are insured against earthquake damage. Four homeowners arc to be selected at random: let X denote the number among the four who have earthquake insurance.

  1. a. Find the probability distribution of X. [Hint: Let S denote a homeowner who has insurance and F one who does not. Then one possible outcome is SFSS, with probability (.25)(.75)(.25)(.25) and associated X value 3. There are 15 other outcomes.]
  2. b. Draw the corresponding probability histogram.
  3. c. What is the most likely value for X?
  4. d. What is the probability that at least two of the four selected have earthquake insurance?

a.

To determine

Find the probability distribution of X.

Explanation of Solution

Given info:

Some parts of California are particularly earthquake-prone. In a metropolitan area, 25% of all homeowners are insured against earthquake damage. Four home owners are selected at random. Let X be the number among the four who have earthquake insurance.

Calculation:

LetS denotes a homeowner who has insurance and F denotes a homeowner who has not insurance.

Thus,P(S)=0.25 and P(F)=0.75.

The possible values of X

X={0,ifnopersonhasinsurance.1,ifonly onepersonhasinsurance2,iftwopersonshaveinsurance3,ifthreepersonshaveinsurance4,ifallpersonshaveinsurance

The sample space is:

{FFFF,FFFS,FFSF,FSFF,SFFF,FFSS,FSFS,FSSF,SFFS,SFSF,SSFF,FSSS,SFSS,SSFS,SSSF,SSSS}

For X=0, the possible outcome is {FFFF}. That is, the frequency is 1. The probability is,

p(x)=(0.75)(0.75)(0.75)(0.75)=(0.75)4

For X=1, the possible outcome is {FFFS,FFSF,FSFF,SFFF}

b.

To determine

Draw the corresponding probability histogram.

c.

To determine

Find the most likely value for X.

d.

To determine

Find the probability that at least two of the four selected have earthquake insurance.

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Probability and Statistics for Engineering and the Sciences
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