A teacher has two large containers filled with blue, red, and green beads and claims that the proportion of red beads is the same in each container. The students believe the proportions are different. Each student selects 50 beads, counts the number of red beads, and returns the beads to the container. The student repeats this process for the second container. One student's samples contained 13 red beads from the first container and 16 red beads from the second container.

Let $ p_1 $ be the true proportion of red beads in container 1 and $ p_2 $ be the true proportion of red beads in container 2. Which of the following is a correct statement for the conditions for this test?

A. The random condition is not met.
B. The 10% condition is not met.
C. The Large Counts Condition is not met.
D. All conditions for inference are met.

Based on the information given, all conditions -- the random condition, the 10% condition, and the large counts condition -- are met for valid statistical inference about the proportions of red beads in the two containers.

Explanation:

The claim that the proportions of red beads are the same in both containers is tested using statistical inference for proportions. In order to make a valid inference, several conditions must be met: the random condition, the 10% condition, and the large counts condition.

Random Condition: This is met if each student's selection of beads is random and independent from one another. Since the student selects beads randomly, returns them and repeats the process, this condition appears to be met.

10% Condition: This condition is met if the sample size is less than 10% of the population. Assuming the containers have a large number of beads, selecting 50 beads should not breach this condition.

Large Counts Condition: For the large counts condition to be met we need np₁ > 5, nq₁ > 5, np₂ > 5, and nq₂ > 5, where n is the sample size, and p and q represent the success and failure probabilities, respectively. With n = 50 and the number of successes being either 13 or 16, it is clear that this condition is also met (as 13 and 16 are both greater than 5).

Therefore, all conditions for inference on the proportions have been met based on the provided information, so the correct choice is D: All conditions for inference are met.