  Let $M(n, k, p)$ denote the maximum probability of the event $X_1 = X_2 =
\cdots = X_n=1$ under a $k$-wise independent distribution whose marginals are
Bernoulli random variables with mean $p$. A long-standing question is to
calculate $M(n, k, p)$ for all values of $n,k,p$. This question has been
partially addressed by several authors, primarily with the goal of answering
asymptotic questions. The present paper focuses on obtaining exact expressions
for this probability. To this end, we provide closed-form formulas of
$M(n,k,p)$ for $p$ near 0 as well as $p$ near 1.
