Final answer:
The question involves determining if the proportion of the class choosing a particular ride as their favorite is the same for two different classes, which cannot be determined from the information provided. The situation described is an example of a voting cycle, highlighting the challenges of decision-making in groups.
Step-by-step explanation:
The question pertains to a situation where a group is trying to determine a single favorite from various options through voting, but encounters a voting cycle. This voting paradox is known as Condorcet's paradox, which illustrates a situation in social choice theory where collective preferences can be cyclic (not transitive) despite the individual preferences being transitive. To address the question 'For which ride was the proportion of the class who chose it as their favorite the same for both classes?', we need to analyze the data provided. Unfortunately, the data necessary to answer this particular question is not present in the information given. However, based on the general concept, if two classes each voted for their favorite ride and the proportion who chose a specific ride (say, mountain biking) as their favorite was identical in both classes, then mountain biking would be the ride with equal proportions.
The explanation provided in the prompt relates to individual and group preferences often found in the field of mathematics called social choice theory. In the hypothetical scenario, students from two classes vote on their favorite activities, and the results create a cyclical pattern where no single option is the definitive favorite amongst the majority when all alternatives are compared pairwise. This result contradicts what one may believe about majority preferences, showcasing a complex decision-making issue where the majority rule doesn't yield a clear winner.