Question

Suppose that \(20 \%\) of all homeowners in an earthquakeprone area of California are insured against earthquake damage. Four homeowners are selected at random. Define the random variable \(x\) as 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 (0.2)(0.8)(0.2)(0.2) 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

For part a, the probability distribution of \(x\) is calculated for every possible value. For part b, the most likely value of \(x\) is the one with the highest probability. For part c, calculate the sum of the probabilities where \(x \geq 2\).

Step-by-step solution

Define the Binomial Distribution

The binomial distribution, denoted \(B(n, p)\), is a discrete probability distribution of the number of successes in a sequence of \(n\) independent experiments. Here, \(n = 4\) and \(p = 20% = 0.2\). The probability of exactly \(k\) successes is given by the formula: \[P(x = k) = C(n, k) * p^k * (1 - p)^{n-k}\] where \(C(n, k)\) is the combination of \(n\) items taken \(k\) at a time.

Calculate the Probability Distribution of \(x\)

To find the probability distribution of \(x\), calculate the probability of \(x\) successes for \(x = 0, 1, 2, 3, 4\). For example, for \(x = 0\) (no homeowners insured), use the formula: \[P(x = 0) = C(4, 0) * 0.2^0 * (1 - 0.2)^{4-0} = 0.41\] Follow the same steps to find the probabilities for \(x = 1, 2, 3, 4\).

Find the Most Likely Value of \(x\)

The most likely value of \(x\) is the one with the highest probability. Compare the probabilities you've found in Step 2 and determine which \(x\) has the highest probability.

Find the Probability of At Least Two Homeowners Being Insured

The probability of at least two homeowners being insured, \(P(x \geq 2)\), is found by summing up the probabilities for \(x = 2, 3, 4\). Calculate as follows: \[P(x \geq 2) = P(x = 2) + P(x = 3) + P(x = 4)\] Use the probabilities calculated in Step 2 to find this sum.