Quantum entanglement is a phenomenon in which a group of particles that are either generated, interacting with each other, or close in proximity to each other, have a property that the quantum states of each particle cannot be described independently of the states of the other particles. This phenomenon was initially investigated by Albert Einstein, Boris Poldosky, and Nathan Rosen in their landmark paper known as the EPR paradox, in which Einstein described this behavior as "spooky action at a distance''. This thesis presents a mathematical and theoretical approach in defining quantum entanglement by detecting photons in their entangled state through a set of photon-number-resolving (PNR) detectors and threshold detectors. This theoretical approach is made rigorous by including the notion of a dark count, a phenomenon in which a detector incidentally detects a photon when it should not have been detected. With this dark count model, we define the probabilities of finding a coincidence of such entangled photons through a combination and configuration of PNR detectors and threshold detectors. Then, we find the coincidence probabilities of detecting a single coincidence of photons within the detector system, the total coincidence probabilities of detecting this coincidence with respect to ground truth, and the effective density matrix that characterizes how well each combination and configuration of detectors can detect photon coincidences. By making mathematical and probabilistic assumptions on the distribution of photon types and counts with respect to ground truth, we are able to compute these quantities and analyze their expressions based on a mathematical and conceptual context.