For each positive integer $n$, what is the largest value of $p$ such that zero is the most likely number of successes in $n$ independent trials with success probability $p$? (Pitman's Probability 2.1.8)
In the solution, it says "if $np + p = 1$, then the mode of bin($n, p$) = $int(np + p$) = $0$ or $1$, both being equally likely." I don't understand why this is true. Isn't $int(1)$ = $1$? Why is $0$ and $1$ both equally likely? Thank you.
When $(n+1)p$ is an integer then the maximum of the binomial mass function occurs at two values: $(n+1)p$ and $(n+1)p-1$. In your case, $(n+1)p=1$.
For more reference, refer to https://en.wikipedia.org/wiki/Binomial_distribution.