= Solution
Let $h(\beta)=\Phi(x_{\mathrm{test}}^T\beta)$. To distinguish the number of posterior draws from the training-sample size, write it as $N$; in the question's sample notation $N=n$. The <Monte Carlo estimator> of the requested <posterior mean> is
$$
\boxed{\widehat\mu_N=\frac1N\sum_{j=1}^N\Phi(x_{\mathrm{test}}^T\beta^{(j)}).}
$$
For independent exact posterior draws it is unbiased, and
$$
\operatorname{Var}(\widehat\mu_N)=\frac1N\operatorname{Var}(h(\beta)\mid Y).
$$
The features are conditioned on as fixed data. The estimated probability concerns a future genuine banknote under the <posterior predictive probability>, not an indicator sampled from that predictive distribution.
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