Intuitive way to arrive at the maximizing argument for the binomial probability

Question

The binomial probability term $q^{n}(1-q)^{N-n}$ is maximized when $q=n/N$. This can be easily arrived at by differentiating the given probability term with respect to q. Is there a more intuitive way to arrive at this value of q that maximizes the probability ?

Answer 1

Yes; $N\choose n$$q^{n}(1-q)^{N-n}$ is the probability of obtaining $n$ successes in $N$ independent trials given a probability of success for each trial of $q$. To maximize the probability of obtaining $n$ successes, choose $q$ such that the expected number of successes in $N$ trials is $n$, i.e., $q = n/N$.

Answer 2

This is how I usually explain the situation:

I repeat some experiment (make up one that makes the most sense) 1000 (or any large number) and find that I get a particular result 50 times. What do you think the probability of the result is?

Almost everyone immediately would say 50/1000.

I then use your result to actually show that this makes the most sense.

I carefully avoid terms like a-posterori or maximum likelihood