= Solution
The two processes are <electron capture> and <beta-plus decay>:
$$
\boxed{{}^{40}_{19}\mathrm K+e^-\to{}^{40}_{18}\mathrm{Ar}+\nu_e,\qquad
{}^{40}_{19}\mathrm K\to{}^{40}_{18}\mathrm{Ar}+e^++\nu_e.}
$$
For neutral atomic <masses>, their available total <energies> obey
$$
Q_{\mathrm{EC}}=M_a(\mathrm K)-M_a(\mathrm{Ar}),\qquad
Q_{\beta^+}=M_a(\mathrm K)-M_a(\mathrm{Ar})-2m_e.
$$
Hence \b[electron capture has approximately $2m_ec^2=1.022\,\mathrm{MeV}$ more available energy]. Electron-hole energies and <nuclear recoil> give smaller corrections for capture from a particular shell. With the physical mass difference, $Q_{\mathrm{EC}}\simeq1.505\,\mathrm{MeV}$ and the positron kinetic-energy budget is only about $0.483\,\mathrm{MeV}$.
<Electron capture> dominates the argon-producing decays. Energetic advantage alone is not a universal theorem about branching fractions: electron density at the nucleus and nuclear transition matrix elements and selection rules also matter. In this case capture can populate the approximately $1.461\,\mathrm{MeV}$ excited argon level, whereas positron emission cannot; this is an additional reason the physical capture branch is much larger.<The KDK experiment by KDK collaboration (2021)>[Dominant capture branch to the 1460.8-keV argon level and rare positron branch]{ref}
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