/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 Suppose that \(20 \%\) of all ho... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 15

Suppose that \(20 \%\) of all homeowners in an earthquake-prone area of California are insured against earthquake damage. Four homeowners are 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 \(\mathrm{F}\) one who does not. Then one possible outcome is SFSS, with probability \((.2)(.8)(.2)(.2)\) and associated \(x\) value of \(3 .\) There are 15 other outcomes.) b. What is the most likely value of \(x\) ? c. What is the probability that at least two of the four selected homeowners have earthquake insurance?

Short Answer

Expert verified
The probability distribution of \(x\) can be calculated using the binomial distribution formula. The most likely value of \(x\) is the one with the highest probability in the distribution. The probability that at least two homeowners have insurance can be calculated by summing up the probabilities of \(x = 2, 3, 4\). Be sure to carry out the steps to get the specific numerical answers.

Step by step solution

01

Find the Probability Distribution of \(x\)

A binomial distribution function can be defined as: \( P(k, n) = C(n, k) * (p^k) * (q^{n-k}) \), where \(n\) is the number of trials, \(k\) is the number of successful trials, \(p\) is the probability of success on a single trial and \(q\) is the probability of failure. In this case, \(n = 4\) (four homeowners selected at random), \(k\) can vary from 0 to 4 denoting the number among the four who have earthquake insurance, \(p = 0.2\) (20% homeowners who have insurance) and \(q = 1 - p = 0.8\). Use this formula to calculate the probability distribution for \(x = 0, 1, 2, 3, 4\).
02

Determine the Most Likely Value of \(x\)

The most likely value of \(x\) is the one that has the highest probability in the probability distribution you calculated in Step 1. So compare the probabilities for each value and choose the one with the highest value.
03

Calculate the Probability That at Least Two Selected Homeowners Have Insurance

The phrase 'at least two of the selected homeowners have earthquake insurance' implies \(x = 2, 3, 4\). In this case, the required probability is given by the sum of the probabilities for these three values of \(x\). Use the probability distribution you calculated in Step 1 and add the probabilities for \(x = 2, 3, 4\).

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