Three randomly selected households are surveyed. The numbers of people in the households are 2,3 , and 10 . Assume that samples of size $n=2$ are randomly selected with replacement from the population of 2,3 , and 10 . Construct a probability distribution table that describes the sampling distribution of the proportion of odd numbers when samples of sizes $n=2$ are randomly selected. Does the mean of the sample proportions equal the proportion of odd numbers in the population? Do the sample proportions target the value of the population proportion? Does the sample proportion make a good estimator of the population proportion? Listed below are the nine possible samples. Construct the probability distribution table. (Type an integer or fraction.)

Step 1 ：Define the population as [2, 3, 10] and the possible samples of size 2 as [(2, 2), (2, 3), (2, 10), (3, 2), (3, 3), (3, 10), (10, 2), (10, 3), (10, 10)].

Step 2 ：Calculate the proportion of odd numbers in each sample. The proportions are [0.0, 0.5, 0.0, 0.5, 1.0, 0.5, 0.0, 0.5, 0.0].

Step 3 ：Calculate the probability of each proportion. The probabilities are [0.4444444444444444, 0.4444444444444444, 0.4444444444444444, 0.4444444444444444, 0.1111111111111111, 0.4444444444444444, 0.4444444444444444, 0.4444444444444444, 0.4444444444444444].

Step 4 ：Construct the probability distribution table as follows: \[ \begin{array}{ccc} \text{Sample} & \text{Proportion of Odd Numbers} & \text{Probability} \\ \hline (2, 2) & 0.0 & 0.444444 \\ (2, 3) & 0.5 & 0.444444 \\ (2, 10) & 0.0 & 0.444444 \\ (3, 2) & 0.5 & 0.444444 \\ (3, 3) & 1.0 & 0.111111 \\ (3, 10) & 0.5 & 0.444444 \\ (10, 2) & 0.0 & 0.444444 \\ (10, 3) & 0.5 & 0.444444 \\ (10, 10) & 0.0 & 0.444444 \\ \end{array} \]

Step 5 ：Calculate the mean of the sample proportions, which is 0.3333333333333333.

Step 6 ：Calculate the proportion of odd numbers in the population, which is 0.3333333333333333.

Step 7 ：Compare the mean of the sample proportions to the population proportion. They are equal, indicating that the sample proportions target the value of the population proportion.

Step 8 ：Conclude that the sample proportion makes a good estimator of the population proportion. \(\boxed{\text{Yes, the mean of the sample proportions equals the proportion of odd numbers in the population. Yes, the sample proportions target the value of the population proportion. Yes, the sample proportion makes a good estimator of the population proportion.}}\)