  While useful probability bounds for $n$ pairwise independent Bernoulli random
variables adding up to at least an integer $k$ have been proposed in the
literature, none of these bounds are tight in general. In this paper, we
provide several results in this direction. Firstly, when $k = 1$, the tightest
upper bound on the probability of the union of $n$ pairwise independent events
is provided in closed-form for any input marginal probability vector
$\mathbf{p} \in [0,1]^n$. To prove the result, we show the existence of a
positively correlated Bernoulli random vector with transformed bivariate
probabilities, which is of independent interest. Building on this, we show that
the ratio of the Boole union bound and the tight pairwise independent bound is
upper bounded by $4/3$ and that the ratio is attained. Applications of the
result in correlation gap analysis and distributionally robust bottleneck
optimization are discussed. The result is extended to find the tightest lower
bound on the probability of the intersection of $n$ pairwise independent
events. Secondly, for any $k \geq 2$ and input marginal probability vector
$\mathbf{p} \in [0,1]^n$, new upper bounds are derived by exploiting ordering
of probabilities. Numerical examples are provided to illustrate when the bounds
provide improvement over existing bounds. Lastly, we identify specific
instances when the existing and the new bounds are tight, for example, with
identical marginal probabilities.
