Say we have $a_1, a_2, ..., a_m \in \{0, 1\}^n$ where the sum over the elements of each vector $a_i$ is $k$.
Let $b \in \{0, 1\}^n$ be a random vector based on the uniform distribution.
Also, let $c_i = a_i^T \cdot b$, which is a scalar value.
$E[c_i] = (1/2) * k = k/2$, since the probability that $b$ has an element '1' at the same index as $a_i$ is (1/2).
However, what is $E[\max_i c_i]$ ? or the highest expected scalar value over all $c_i$ values for $i=\{1,2,...,m\}$.
Clearly, $E[\max_i c_i]$ should be closer to $k$ and should be a function of $m$.