This is very useful for computing by recursion the probability mass of the binomial.

Property:

For *X* a B(n,p) random variable
with probability of success *p* neither 0 or 1, then as *k* varies
from 0 to *n*,
*P*(*X*=*k*) first increases monotonically
and then decreases monotonically,
(it is unimodal)
reaching its highest value when *k* is the largest
integer less or equal to
(*n*+1)*p* (=*floor*(*n*+1)*p*).

Proof:

is equivalent to

The value where the the probability mass function takes on its maximum is called the mode.