= Solution
Equality can occur only at the two pairs $(0,0)$ and $(1,1)$. Their combined <probability> is
$$
e^{-p}-p(1-e^{-p})+pe^{-p}=(1+2p)e^{-p}-p.
$$
Taking the complementary <probability> gives
$$
\boxed{\mathbb P(X\ne Y)=1+p-(1+2p)e^{-p}.}
$$
Since $1+2p>0$, the supplied exponential bound implies
$$
\mathbb P(X\ne Y)\leqslant1+p-(1+2p)(1-p)=\boxed{2p^2}.
$$
This is a bound on the mismatch <probability> for this particular <coupling> of the two <marginal distributions>; the variables are not presumed independent.
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