Final answer:
This is a binomial probability problem where we calculate the likelihood of a certain number of patients waiting longer than the median time in an emergency room. It involves summing the probabilities of different numbers of patients within the specified ranges, an approximation or computational tool is needed to find exact probabilities.
Step-by-step explanation:
The question belongs to the field of probability and involves calculating the chances of an event occurring, based on given statistics. We need to determine the likelihood that a certain number of patients will have to wait longer than the median wait time in an emergency room given that the median wait time is 30 minutes. This is a typical problem that can be approached using the binomial distribution, as we have a fixed number of trials (200 patients), two outcomes (waiting less than or more than 30 minutes), and a constant probability for each trial (since the median wait time is 30 minutes, the probability of waiting more than 30 minutes is 0.5).
To find the probability of more than half of the patients (i.e. more than 100 patients) waiting for more than 30 minutes, we need to consider the sum of probabilities for all outcomes where 101 to 200 patients wait for more than 30 minutes. Similarly, for the other parts, we have to calculate the probabilities for the given numbers of patients (more than 110, and more than 80 but less than 120).
Since these calculations involve summation of many terms of a binomial distribution, using a normal approximation or a binomial probability calculator would be practical. Unfortunately, without specific calculations or software tools, we cannot give the exact probabilities.