I just got a problem related to binomial distribution as following.
Which value of $k$ makes $\left(\begin{matrix}n \\ k\end{matrix} \right)p^k(1-p)^{n-k}$ as large as possible?
I've spend hours on this question but have no idea how to calculate the derivative for factorial. Thank you in advance!
Let the terms be $t_k = {n \choose k } p^k (1-p)^{n - k}$.
Hint: Consider the ratio $ \frac{ t_k} { t_{k-1} } $. When is this more than 1? When is this less than 1?