Theoretical Median and Mean question How likely is this scenario?

In a class of 100 people who take an exam, the median of the exam is 82 and the mean is 77.
  What if the sample size is cut in half and the median/mean stay the same?

So how likely is the initial scenario and how does it change when the sample size is reduced or increased.
Thanks for any help.
Edit: I would think that a difference between the median and mean for a sample of that size would be unlikely in the first place?
 A: Suppose $X$ is a random variable which denotes the marks in the exam. Given a sample $\{X_1, X_2, \ldots , X_n\}$ of size $n$, the mean and median are both well known measures of central tendency of the distribution of $X$. Since it is random variables we're talking about, and the sample mean and sample median are actually realizations of a random process, we do not have a hold on the exact values of these quantities. We can only handle them in terms of chance, i.e. probability. As $n$ increases, we expect the mean and median to get closer. But in reality, they only get closer in terms of chance, i.e. for some fixed pre-specified positive real number $\epsilon > 0$,

$$P_n(\epsilon)=Pr(|\overline{X_n}-X_n^{med}|>\epsilon)$$ 

becomes smaller as $n$ increases (Of course the quantity will depend on the underlying distribution of $X$). So even if you consider a sample of size $10000$ instead of one of size $100$, it is still possible that the realized value of sample mean and sample median differs a lot. But the probability that they will differ more than some pre-specified quantity will be smaller. That said, there are some subtle features that really make these measures different. For example, median is a robust estimator, while mean is not. If you change a sample value and make it very large or very small, the mean will be affected badly, but the median remains same, and hence the median is said to be more stable than the mean. Hope that helps.
A: I think this question is tougher than it seems at first glance. However, if you randomly sample a population, you would not change the theoretical mean or median. That can be proved with a little technical setup.
You mentioned though that this is a finite class with 100 students and not a theoretically infinite population. Therefore, the sample mean and median are estimators which would likely have non-neglible variances. If the sample mean and median for the whole class is 82 and 77, then recalculating these estimators on half the class would almost certainly alter their values.
*I'm new here so I can't comment directly on BruceET's answer. But I'll add that the simulation he did is highly dependent on the distribution of the underlying data. Thus, speculation is all that we can really do here. To emphasize this point, consider what would we redefined the original R data as
x = c(rep(44.5102, 49), rep(78, 2), rep(100, 49))
Then, the mean and median are the if we run the same simulation, the mean will only be less than the median 49% of the time -- not 100%.
