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The $\mu$Dose system was developed to allow the measurement of environmentallevels of natural radioactive isotopes. The system records $\alpha$ and $\beta$particles along with four decay pairs arising from subsequent decays of$^{214}$Bi/$^{214}$Po, $^{220}$Rn/$^{216}$Po, $^{212}$Bi/$^{212}$Po and$^{219}$Rn/$^{215}$Po. Under the assumption of secular equilibrium this allowsto assess the specific radioactivities of $^{238}$U, $^{235}$U, $^{232}$Thdecay chains and $^{40}$K. This assessment provides results with uncertaintieswhich are correlated and, thus, require the development of an error estimationmethodology which considers this issue. Here we present two differentapproaches for uncertainty propagation based on Monte Carlo and Bayesianmethods. Both approaches produce statistically indistinguishable results andallow significantly better dose rate precision than when the correlations arenot accounted for. In the given example, the dose rate precision is improved bya factor of two.

Increased dose rate precision in combined $\alpha$ and $\beta$ counting in the $\mu$Dose system - a probabilistic approach to data analysis