A Josephson radiation threshold detector (JRTD) that is based on the threshold behaviour of a current bias Josephson junction (CBJJ) is designed and fabricated for infrared radiation (IR@1550nm) detection at low temperatures. To achieve the optimal performance, we develop a binary hypothesis detection method to calibrate Josephson threshold behaviours (i.e. the switching current distributions of the CBJJ with the Al/AlO_x/Al junction) in the absence and presence of radiation. In the absence of IR radiation, the junction transitioned with a measurable voltage drop across the junction, and this signal was treated as the events of hypothesis H₀. The events of junction transition observed in the presence of the IR radiation served as hypothesis H₁. Considering the usual Gaussian noise and based on statistical decision theory, the accumulated data of the measured switching current distributions are processed, and the threshold sensitivity of the demonstrated JRTD device is estimated. The minimum detectable IR radiation power of the proposed detector is approximately 0.74 pW, which corresponds to the photon rate of 5.692 × 10⁶ photons/second. Further optimisation of JRTDs to implement the desired binary detection of a single photon is still a subject of argument, at least theoretically.