= Solution
\b[The ideal two-shower event fraction is $B_e^2$, approximately $4\%$ in simple channel counting or $3.18\%$ using the measured <tau lepton> electronic <branching fraction>.] The selected <particle decay> chain is
$$
\tau^-\to e^-\bar\nu_e\nu_\tau,\qquad
\tau^+\to e^+\nu_e\bar\nu_\tau.
$$
The <electron> and <Positron> each make a contained <electromagnetic shower>; <muons> penetrate, and generic <hadrons> do not deposit all their <energy> as a contained <electromagnetic shower>.
To estimate $B_e$, use <lepton universality> and neglect final <masses> and <Quantum chromodynamics> corrections. The electronic and muonic <decay widths> each carry weight one. The accessible inclusive hadronic channels have weights $3|V_{ud}|^2$ and $3|V_{us}|^2$, where three is the <color multiplicity in a decay width>. The <charm quark> is inaccessible in physical <tau lepton> hadronic decays. With $|V_{ud}|^2+|V_{us}|^2\simeq1$, the total weight is $1+1+3=5$, giving
$$
B_e\simeq\frac15,\qquad
\boxed{\frac{N_{\rm two\ showers}}{N_{\tau^+\tau^-}}=B_e^2\simeq\frac1{25}.}
$$
The 2021 measured value $B_e=0.1782$ instead gives $B_e^2=0.0317552$.<Review of Particle Physics by Particle Data Group (2021)>[Tau listing, p. 6, mode 5: electronic branching fraction $(17.82\pm0.04)\%$]{ref} The difference reflects effects omitted by the simple free-channel estimate, especially the hadronic contribution to the total <decay width>.
There are two <tau leptons> per produced pair and two per selected pair, so the fraction of all produced <tau leptons> belonging to these two-shower events is also $B_e^2$. The unrestricted fraction of individual <tau leptons> decaying electronically is $B_e$; these are different selections. A real high-<energy> shower cut gives an additional acceptance factor: $N_{\rm selected}/N_{\rm pairs}=B_e^2\varepsilon_{ee}$, which cannot be computed without the cut and detector response. No factor of two is needed for the unique $e^+e^-$ assignment.
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