Final answer:
The question deals with using mathematical concepts to understand and predict election outcomes. It involves calculations to determine probabilities from given population proportions and sample survey results, as well as the implications of having multiple candidates in a political race.
Step-by-step explanation:
The question presented appears to be asking about electoral dynamics and probabilities related to elections, which falls within the realm of Mathematics, specifically statistics and probability. The question also has a political aspect but its core is focused on mathematical analysis. The reference provided suggests a scenario where the probability is required to find the chance of a mayor winning an election based on sample survey results and population preferences.
For example, to find the probability that at least 250 out of 500 people favor the incumbent mayor, Dawn Morgan, we use her support rate (46%) and apply statistical methods, such as the use of a continuity correction factor for approximating a binomial distribution with a normal distribution, as the sample size is large. This calculation would require computing the z-score for the sample proportion and then using the standard normal distribution to find the desired probability.
In a scenario where the electorate is split 60-40 between liberals and conservatives, the introduction of a third candidate could indeed split the vote, potentially altering the expected outcome based on initial population proportions. This demonstrates how electoral outcomes can be affected by the number of candidates and their political alignments.