Question

students in a class were surveyed about the number of children in their families. the results of the survey are shown in the table.

number of children in family	number of surveys
two	18
three	22
four	8
five or more	3

two surveys are chosen at random from the group of surveys. after the first survey is chosen, it is returned to the stack and can be chosen a second time. what is the probability that the first survey chosen indicates four children in the family and the second survey indicates one child in the family?

• $\frac{1}{50}$
• $\frac{2}{15}$
• $\frac{3}{20}$
• $\frac{17}{60}$

Explanation:

Step1: Calculate total number of surveys

First, sum up the number of surveys for each category. The number of surveys for one, two, three, four, five or more children are 9, 18, 22, 8, 3 respectively. So total number of surveys \( n = 9 + 18 + 22 + 8 + 3 \).
\[ n=9 + 18+22 + 8+3=60 \]

Step2: Find probability of first event (four children)

The number of surveys with four children is 8. So the probability \( P(\text{four})=\frac{\text{Number of four - child surveys}}{\text{Total number of surveys}}=\frac{8}{60}=\frac{2}{15} \) (simplified). But wait, actually we can keep it as \( \frac{8}{60} \) for now.

Step3: Find probability of second event (one child)

Since the first survey is returned, the total number of surveys is still 60. The number of surveys with one child is 9. So the probability \( P(\text{one})=\frac{\text{Number of one - child surveys}}{\text{Total number of surveys}}=\frac{9}{60}=\frac{3}{20} \) (simplified). Or keep as \( \frac{9}{60} \).

Step4: Calculate the combined probability

Since the two events are independent (because the first survey is returned), the probability of both events happening is the product of their individual probabilities. So \( P = P(\text{four})\times P(\text{one})=\frac{8}{60}\times\frac{9}{60} \)? Wait, no, wait. Wait, no, the total number of surveys is 60. The number of four - child surveys is 8, one - child is 9. So the probability that first is four and second is one is \( \frac{8}{60}\times\frac{9}{60} \)? No, wait, no. Wait, the total number of surveys is 60. So the probability of choosing a four - child survey first is \( \frac{8}{60} \), and then choosing a one - child survey is \( \frac{9}{60} \) (since we put the first one back). So \( P=\frac{8}{60}\times\frac{9}{60} \)? Wait, no, that's wrong. Wait, no, the total number of surveys is 60. So the number of four - child surveys is 8, so \( P(\text{first is four})=\frac{8}{60} \), and \( P(\text{second is one})=\frac{9}{60} \). Then the combined probability is \( \frac{8}{60}\times\frac{9}{60} \)? Wait, no, wait, no. Wait, the total number of surveys is 60. Let's recalculate total: 9 + 18+22 + 8+3 = 60. So number of four - child surveys: 8, one - child: 9. So the probability that first is four (8 out of 60) and second is one (9 out of 60, since replaced) is \( \frac{8}{60}\times\frac{9}{60} \)? Wait, no, that can't be. Wait, no, I made a mistake. Wait, the problem is: first survey is four children, second is one child. So the number of four - child surveys is 8, total surveys 60. So \( P(\text{four})=\frac{8}{60} \). Then, since we return the first survey, the total is still 60, and the number of one - child surveys is 9, so \( P(\text{one})=\frac{9}{60} \). Then the combined probability is \( \frac{8}{60}\times\frac{9}{60} \)? Wait, no, that's not matching the options. Wait, maybe I miscalculated the total. Wait 9 (one) + 18 (two)+22 (three)+8 (four)+3 (five or more) = 9 + 18 is 27, 27+22 is 49, 49 + 8 is 57, 57+3 is 60. Correct. So 8 (four) and 9 (one). So \( \frac{8}{60}\times\frac{9}{60}=\frac{72}{3600}=\frac{1}{50} \)? Wait, \( \frac{8\times9}{60\times60}=\frac{72}{3600}=\frac{1}{50} \). Wait, but let's check the options. One of the options is \( \frac{1}{50} \). Wait, but let's re - do the calculation. Wait, the number of four - child surveys is 8, total is 60. So probability of four is \( \frac{8}{60} \). Probability of one is \( \frac{9}{60} \). Then multiply them: \( \frac{8}{60}\times\frac{9}{60}=\frac{72}{3600}=\frac{1}{50} \). Wait, but let's check the arithmetic again. 8 times 9 is 72, 60 times 60 is 3600. 72 divided by 3600: divide numerator and denominator by 72, we get \( \frac{1}{50} \). Yes.

Wait, but let's check the steps again. Total number of surveys: 9 + 18+22 + 8+3 = 60. Number of four - child surveys: 8. Number of one - child surveys: 9. Since the first survey is returned, the two events are independent. So \( P=\frac{8}{60}\times\frac{9}{60}=\frac{72}{3600}=\frac{1}{50} \).

Answer:

\( \boldsymbol{\frac{1}{50}} \) (corresponding to the option \( \frac{1}{50} \))