Conditional Probability on a Six-Position Circular Arrangement

A circular device has six positions, exactly two marked and four unmarked. A starting position is chosen uniformly at random. You observe that the current position is unmarked. What is the probability that the next position, advancing one step around the circle, is marked? Compare three cases: the two marked positions are uniformly selected from all pairs; the pair is known to be nonadjacent; and the pair is known to be adjacent. Adjacency includes the first and last positions around the circle. ### Clarifying Questions Use one-step advancement without choosing a new random starting position between observations. In each restricted case, assume uniform selection among allowed pairs, independently of the initial starting position. This abstract marked-position formulation preserves the conditional-probability task. ### What a Strong Answer Covers Condition on the observed unmarked position and count valid next-position events for each placement rule. Distinguish advancement from independently randomizing the next position. ### Follow-up Questions - What happens if a fresh position is selected uniformly before the second observation? - Why do adjacency restrictions change the conditional probability even though the number of marked positions remains two?