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{2}{25}\\)

\\(\frac{17}{50}\\)

Explanation:

Step1: Calculate total number of surveys

First, we sum up the number of surveys for each category. So, \(9 + 18 + 22 + 8 + 3 = 60\)? Wait, no, wait: 9 (one) + 18 (two) + 22 (three) + 8 (four) + 3 (five or more) = 9+18=27, 27+22=49, 49+8=57, 57+3=60? Wait, the problem's options have denominators like 50, 15, 25, 75. Wait, maybe I miscalculated. Wait, let's check again: one:9, two:18, three:22, four:8, five or more:3. 9+18=27, 27+22=49, 49+8=57, 57+3=60. Wait, but the options have 50, 15, 25, 75. Maybe a typo? Wait, maybe the table is different. Wait, maybe the "four" is 8, "one" is 9, "two" 18, "three"22, "five or more"3. Wait, maybe the total is 9+18+22+8+3=60. But the first event is "four children", which has 8 surveys, and the second is "one child", which has 9 surveys. Since it's with replacement, the probability is \(P(\text{first four}) \times P(\text{second one})\). So \(P(\text{four}) = \frac{8}{60}\), \(P(\text{one}) = \frac{9}{60}\). Then multiply: \(\frac{8}{60} \times \frac{9}{60} = \frac{72}{3600} = \frac{1}{50}\)? Wait, 89=72, 6060=3600, 72/3600=1/50. Oh, maybe the total is 60? Wait, 9+18+22+8+3=60. So first, probability of four children: number of four-children surveys is 8, total is 60. Probability of one child: 9/60. So multiply: (8/60)(9/60) = (89)/(6060) = 72/3600 = 1/50. So that's the first option. Wait, but let's check again. Wait, maybe the total is 60. So step1: find total surveys: 9+18+22+8+3=60. Step2: probability first is four: 8/60. Step3: probability second is one: 9/60 (since replaced). Step4: multiply them: (8/60)(9/60) = 72/3600 = 1/50.

Step2: Calculate probability of first event (four children)

Number of surveys with four children is 8, total surveys is 60. So \(P(\text{four}) = \frac{8}{60}\).

Step3: Calculate probability of second event (one child)

Number of surveys with one child is 9, total surveys is 60 (since replaced). So \(P(\text{one}) = \frac{9}{60}\).

Step4: Multiply the two probabilities

Since the events are independent (with replacement), we multiply the probabilities: \(\frac{8}{60} \times \frac{9}{60} = \frac{72}{3600} = \frac{1}{50}\).

Answer:

\(\frac{1}{50}\) (the first option, assuming the options are like A. \(\frac{1}{50}\), B. \(\frac{2}{15}\), etc.)