I'm trying to answer a textbook question that asks: in a binomial distribution with $n$ trials and probability of success $p$, what value of $k$ successes has the maximum probability?
What I tried is the following ratio:
$$\frac{{n\choose k}p^k(1-p)^{n-k}}{{n\choose k-1}p^{k-1}(1-p)^{n-k+1}}$$
Where do I go from here? I'm not sure how to simplify this. Any help is appreciated.
Hint: You can simplify the fractions quite a bit. Powers of $p$ can be simplified top and bottom; as can powers of $(1-p)$. Also if you write out the binomial coefficients using their factorial definitions, there will be a lot of simplifying with them also.