  Consider a testing problem for the null hypothesis $H_0:\theta\in\Theta_0$.
The standard frequentist practice is to reject the null hypothesis when the
p-value is smaller than a threshold value $\alpha$, usually 0.05. We ask the
question how many of the null hypotheses a frequentist rejects are actually
true. Precisely, we look at the Bayesian false discovery rate
$\delta_n=P_g(\theta\in\Theta_0|p-value<\alpha)$ under a proper prior density
$g(\theta)$. This depends on the prior $g$, the sample size $n$, the threshold
value $\alpha$ as well as the choice of the test statistic. We show that the
Benjamini--Hochberg FDR in fact converges to $\delta_n$ almost surely under $g$
for any fixed $n$. For one-sided null hypotheses, we derive a third order
asymptotic expansion for $\delta_n$ in the continuous exponential family when
the test statistic is the MLE and in the location family when the test
statistic is the sample median. We also briefly mention the expansion in the
uniform family when the test statistic is the MLE. The expansions are derived
by putting together Edgeworth expansions for the CDF, Cornish--Fisher
expansions for the quantile function and various Taylor expansions. Numerical
results show that the expansions are very accurate even for a small value of
$n$ (e.g., $n=10$). We make many useful conclusions from these expansions, and
specifically that the frequentist is not prone to false discoveries except when
the prior $g$ is too spiky. The results are illustrated by many examples.
