If $X$ is a binomial random variable with parameters $(n, p)$, where $0 < p < 1$, then as $k$ goes from $0$ to $n$, $P\{X = k\}$ first increases monotonically and then decreases monotonically, reaching its largest value when $k$ is the largest integer less than or equal to $(n + 1)p$.
I am trying to understand this proposition, it is not clear the meaning not the proof. The proof of this, it's just considering $P\{X = k\}/P\{X = k − 1\}$ and determining for what values of k it is greater or less than 1.
My question is how is this applicable in real life?, How to well understand this proposition?. Is it just a rate to compare probabilities?
Please, if someone could give me your opinions, I will appreciate it.
Thanks.