To find the probability that the first survey chosen indicates four children in the family and the second survey indicates one child in the family, we can break it down into a few steps:
1. Calculate the total number of surveys:
The total number of surveys is the sum of all the surveys given in the table.
[tex]\[
\text{Total surveys} = 9 + 18 + 22 + 8 + 3 = 60
\][/tex]
2. Calculate the probability of choosing a survey indicating four children in the family (first choice):
The number of surveys indicating four children is 8.
[tex]\[
\text{Probability (first choice, four children)} = \frac{8}{60} = \frac{2}{15} \approx 0.133
\][/tex]
3. Calculate the probability of choosing a survey indicating one child in the family (second choice):
Since the survey is replaced after the first selection, the total number of surveys remains the same (60). The number of surveys indicating one child is 9.
[tex]\[
\text{Probability (second choice, one child)} = \frac{9}{60} = \frac{3}{20} = 0.15
\][/tex]
4. Multiply the probabilities of these two independent events:
Since each choice is independent (the survey is replaced), we multiply the probabilities of the two events.
[tex]\[
\text{Probability (first four children, then one child)} = \left(\frac{8}{60}\right) \times \left(\frac{9}{60}\right) = \frac{2}{15} \times \frac{3}{20} = \frac{6}{300} = \frac{1}{50} = 0.02
\][/tex]
So, the probability that the first survey chosen indicates four children in the family and the second survey indicates one child in the family is [tex]\(\frac{1}{50}\)[/tex], which corresponds to approximately [tex]\(0.02\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{1}{50}}
\][/tex]
