= Solution
Substituting the standardized deviation into the <Bayes factor> approximation gives
$$
\boxed{B_{01}\approx\sqrt{\frac{20000}{\pi}}e^{-4.5}\approx0.89,\qquad B_{10}\approx1.13.}
$$
This is only a slight preference for the alternative, despite the small <p-value>. The two procedures measure different things: the <p-value> sums a tail under the <null hypothesis>, while the <Bayes factor> compares probabilities of the observed count under both models. The uniform alternative <prior distribution> spreads its probability over many success probabilities that fit the observations poorly. Only a narrow range around $0.515$ has large <likelihood>, and integrating over the whole alternative penalizes this diffuse prediction.
The prefactor makes the distinction especially clear. At a fixed standardized deviation $z$, the <p-value> stays approximately $2\{1-\Phi(|z|)\}$, whereas $B_{01}\approx\sqrt{2n/\pi}e^{-z^2/2}$ eventually grows with $n$. A fixed diffuse alternative can consequently favour the point null even while a conventional test rejects it. This is the same prior-width mechanism illustrated by the <Jeffreys-Lindley paradox for a Gaussian point null>.
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