I made a histogram based on a binomial distribution a simple "n pick k" problem. "A bag is filled with red and yellow balls. 10% of the balls are red. If you pull 10 balls at random, what is the probability that exactly 1 ball of the 10 is red?" According to the binomial distribution function, the probability is 38.74% (tell me if my math is wrong).
I made a histogram of various distributions, matching the probability to the number of red balls picked. 10% and 1 red ball picked; 20% and 2 red balls picked, and so on. And I found an interesting behavior. Even though the percentage of red balls in the bag (population) matches the expected percentage of red balls picked (sample), the probabilities of getting the expected number of balls varied. See below:
10%, get 1 reds. 38.74% probability.
20%, get 2 reds. 30.20% probability.
30%, get 3 reds. 26.69% probability.
40%, get 4 reds. 25.08% probability.
50%, get 5 reds. 24.60% probability.
60%, get 6 reds. 25.08% probability.
70%, get 7 reds. 26.69% probability.
80%, get 8 reds. 30.20% probability.
90%, get 9 reds. 38.74% probability.
This looks like a normal distribution turned upside down. It dips in the middle and goes up at the edges.
I don't understand this behavior. There are far many more combinations of balls in the middle of the distribution (ie, 45 combinations with 2 reds vs. 252 combinations with 5 reds), I would expect the probabilities to increase slightly in the middle range. Instead they decrease. Why?
More importantly, is
You're not looking at a distribution, you're looking at a particular rule for deducing one probability for each distribution in a series of distributions.
You could look at it this way: what if $0\%$ of the balls in the bag are red? What if $100\%$ of the balls in the bag are red?
In those two cases, the probability that the number of red balls in the sample will exactly match the expected value is $100\%.$
For $50\%$ red balls in the bag, obviously you have a less than $100\%$ chance to draw a sample in which exactly $50\%$ of the balls are red. So this particular (and actually rather peculiar) property you have defined, "probability that the sample mean is exactly equal to the population mean, as a function of the population mean," must be larger for extreme values of the population mean than for more in-between values.