TITLE: Intuitive way to arrive at the maximizing argument for the binomial probability
QUESTION [8 upvotes]: 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 ?

REPLY [5 votes]: 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., $q = n/N$.