Let
A
1
,
A
2
,
…
,
A
n
{A_1},{A_2}, \ldots ,{A_n}
and
B
1
,
B
2
,
…
,
B
m
{B_1},{B_2}, \ldots ,{B_m}
be two sets of events on a probability space. Let
X
n
{X_n}
and
Y
m
{Y_m}
be the number of those
A
j
{A_j}
and
B
s
{B_s}
, respectively, that occur. Let
S
k
,
t
{S_{k,t}}
be the
(
k
,
t
)
th
(k,t){\text {th}}
binomial moment of the vector
(
X
n
,
Y
m
)
({X_n},{Y_m})
. We establish optimal bounds on
P
(
X
n
⩾
1
,
Y
m
⩾
1
)
P({X_n} \geqslant 1,{Y_m} \geqslant 1)
by means of linear combinations of
S
1
,
1
,
S
2
,
1
,
S
1
,
2
{S_{1,1}},\;{S_{2,1}},\;{S_{1,2}}
and
S
2
,
2
{S_{2,2}}
. Optimal lower bounds are also determined when only
S
1
,
1
,
S
2
,
1
{S_{1,1}},\;{S_{2,1}}
and
S
1
,
2
{S_{1,2}}
are utilized.