Chapter 7.2, Problem 15E

Suppose that 20% of all homeowners in an earthquake-prone area of California are insured against earthquake damage. Four homeowners are selected at random. Suppose that x denotes the number among the four who have earthquake insurance.

1. a. Find the probability distribution of x. (Hint: Let S represent a homeowner who has insurance and F one who does not. Then one possible outcome is SPSS, with probability (0.2)(0.8)(0.2)(0.2) and associated x value of 3. There are 15 other outcomes.)
2. b. What is the most likely value of x?
3. c. What is the probability that at least two of the four selected homeowners have earthquake insurance?

To determine

Obtain the probability distribution for the random variable x.

Answer to Problem 15E

The probability distribution for the random variable x is given below:

x	0	1	2	3	4
P(x)	0.4096	0.4096	0.1536	0.0256	0.0016

Explanation of Solution

Calculation:

The given information is that 20% of all homeowners insured against earthquake damage who live in an earthquake-prone area.

Define the random variable x as the number of homeowners who have earthquake insurance. Four homeowners are randomly selected.

Denote S as the homeowner who has insured and F as the homeowner who did not insure.

Thus, P(S) = 0.2 and P(F) = 0.8

The possible outcomes and the respective probabilities are obtained using the multiplication rule for independent events.

Outcome	Probability	x value
SSSS	$\left(0.2\right)\left(0.2\right)\left(0.2\right)\left(0.2\right)=\text{0}\text{.0016}$	4
SSSF	$\left(0.2\right)\left(0.2\right)\left(0.2\right)\left(0.8\right)=\text{0}\text{.0064}$	3
SSFS	$\left(0.2\right)\left(0.2\right)\left(0.8\right)\left(0.2\right)=\text{0}\text{.0064}$	3
SSFF	$\left(0.2\right)\left(0.2\right)\left(0.8\right)\left(0.8\right)=\text{0}\text{.0256}$	2
SFSS	$\left(0.2\right)\left(0.8\right)\left(0.2\right)\left(0.2\right)=\text{0}\text{.0064}$	3
SFSF	$\left(0.2\right)\left(0.8\right)\left(0.2\right)\left(0.8\right)=\text{0}\text{.0256}$	2
SFFS	$\left(0.2\right)\left(0.8\right)\left(0.8\right)\left(0.2\right)=\text{0}\text{.0256}$	2
SFFF	$\left(0.2\right)\left(0.8\right)\left(0.8\right)\left(0.8\right)=\text{0}\text{.1024}$	1
FSSS	$\left(0.8\right)\left(0.2\right)\left(0.2\right)\left(0.2\right)=\text{0}\text{.0064}$	3
FSSF	$\left(0.8\right)\left(0.2\right)\left(0.2\right)\left(0.8\right)=\text{0}\text{.0256}$	2
FSFS	$\left(0.8\right)\left(0.2\right)\left(0.8\right)\left(0.2\right)=\text{0}\text{.0256}$	2
FSFF	$\left(0.8\right)\left(0.2\right)\left(0.8\right)\left(0.8\right)=\text{0}\text{.1024}$	1
FFSS	$\left(0.8\right)\left(0.8\right)\left(0.2\right)\left(0.2\right)=\text{0}\text{.0256}$	2
FFSF	$\left(0.8\right)\left(0.8\right)\left(0.2\right)\left(0.8\right)=\text{0}\text{.1024}$	1
FFFS	$\left(0.8\right)\left(0.8\right)\left(0.8\right)\left(0.2\right)=\text{0}\text{.1024}$	1
FFFF	$\left(0.8\right)\left(0.8\right)\left(0.8\right)\left(0.8\right)=\text{0}\text{.4096}$	0

The probability distribution for the random variable x is given below:

x value	Probability
0	0.4096
1	$4\times \text{0}\text{.1024}=\text{0}\text{.4096}$
2	$6\times 0.0256=\text{0}\text{.1536}$
3	$4\times 0.0064=\text{0}\text{.0256}$
4	0.0016

To determine

Identify the most likely value of x.

Answer to Problem 15E

The most likely values of x are 0 and 1.

Explanation of Solution

Calculation:

From part (a), the highest probability is 0.4096. This probability is obtained for x = 0 and x = 1.

Thus, the most likely values of x are 0 and 1.

To determine

Obtain the probability that at least two homeowners have earthquake insurance.

Answer to Problem 15E

The probability that at least two homeowners have earthquake insurance is 0.1808.

Explanation of Solution

Calculation:

$\begin{array}{c}P\left(x\ge 2\right)=p\left(2\right)+p\left(3\right)+p\left(4\right)\\ =0.1536+0.0256+0.0016\\ =\text{0}\text{.1808}\end{array}$

Thus, the probability that at least two homeowners have earthquake insurance is 0.1808.