= Solution
Let $\phi_\rho$ denote the centered unit-variance <bivariate normal distribution> density,
$$
\phi_\rho(x,y)=\frac1{2\pi\sqrt{1-\rho^2}}\exp\left(-\frac{x^2-2\rho xy+y^2}{2(1-\rho^2)}\right).
$$
The complement of rejecting either <null hypothesis> is $W_1\le c,W_2\le c$. Therefore \b[the probability of at least one rejection] is
$$
\boxed{1-\int_{-\infty}^{c-m_1}\int_{-\infty}^{c-m_2}\phi_\rho(x,y)\,dy\,dx.}
$$
When both <null hypotheses> are true, this is the <familywise error rate>. If one <null hypothesis> is false, a rejection of that hypothesis is not a <type I error>, so the displayed probability is then an any-rejection probability rather than the <familywise error rate>.
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