To find the maximum variance of a Bernoulli RV first find the variance equation.
$ Var(X) = E[X^2] - (E[X])^2 \! $
We know that for Bernoulli RVs
$ E[X^2] = p \! $
$ E[X] = p \! $
Plug these values into the variance equation, differentiate with respect to p, set equal to 0 and find the value of p that results in the largest value for the variance.
