= Solution
Bayes' formula divides the point-null contribution to the mixture density by the full marginal density:
$$
q(\overline x)
=\frac{\phi_{1/n}(\overline x)}
{\phi_{1/n}(\overline x)+
\phi_{\tau^2+1/n}(\overline x)}.
$$
Since
$$
\frac{\phi_{\tau^2+1/n}(\overline x)}
{\phi_{1/n}(\overline x)}
=\frac1{\sqrt{1+n\tau^2}}
\exp\left[
\frac{n^2\tau^2\overline x^2}{2(1+n\tau^2)}
\right],
$$
the <posterior probability of a Gaussian point null> is
$$
\boxed{
q(\overline x)=
\left\{
1+\frac1{\sqrt{1+n\tau^2}}
\exp\left[
\frac{n^2\tau^2\overline x^2}{2(1+n\tau^2)}
\right]
\right\}^{-1}}.
$$
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