Question

Suppose that $20,000$ married adults in the United States were randomly surveyed as to the number of children they have. The results are compiled and are used as theoretical probabilities. Let X = the number of children married people have.

a. Find the probability that a married adult has three children.

b. In words, what does the expected value in this example represent?

c. Find the expected value.

d. Is it more likely that a married adult will have two to three children or four to six children? How do you know?

Short answer

(a) The probability that a married adult has three children.

(b) The expected value represents the average number of children a married people having.

(c) The expected value represents the average number of children a married people having.

(d) It more likely that a married adult will have two to three children.

Step-by-step solution

Given information (part a)

Given the results of a survey conducted on the number of children married adults have.

Explanation (part a)

$X$	$P\left(X\right)$
$0$	$0.10$
$1$	$0.20$
$2$	$0.30$
$3$	$0.20$
$4$	$0.10$
$5$	$0.05$
$6$ or more	$0.05$
Total	$1$

Because the sum of all probabilities for a random variable is 1, the probability x=3 equals 1-sum of all probabilities.

Given information (part b)

Given the results of a survey conducted on the number of children married adults have.

Explanation (part b)

The expected value in the scenario in question represents the average number of children a married couple has.

Given information (part c)

Given the results of a survey conducted on the number of children married adults have.

Explanation (part c)

$X$	$P\left(X\right)$	$X.P\left(x\right)$
$0$	$0.10$	$0$
$1$	$0.20$	$0.20$
$2$	$0.30$	$0.60$
$3$	$0.20$	$0.60$
$4$	$0.10$	$0.40$
$5$	$0.05$	$0.25$
$6$ or more	$0.05$	$0.30$
Total	$1$	$2.35$

$$\begin{gathered} E\left(x\right)=\underset{}{\sum \left[x.P\left(x\right)\right]} \\ E\left(x\right)=2.35 \end{gathered}$$

Given information (part d)

Given the results of a survey conducted on the number of children married adults have.

Explanation (part d)

A married adult with two to three children is more likely to have two to three children than a married adult with four to six children because the likelihood of having two to three children is greater than the probability of having four to six children. A person's chances of having two to three children are 0.50, while his or her chances of having four to six children are 0.20.