The assessment of fingermark (latent fingerprint) quality is an intrinsic part of a forensic investigation. The fingermark quality indicates the value and utility of the trace evidence recovered from the crime scene in the course of a forensic investigation; it determines how the evidence will be processed, and it correlates with the probability of finding a corresponding fingerprint in the reference dataset. The deposition of fingermarks on random surfaces occurs spontaneously in an uncontrolled fashion, which introduces imperfections to the resulting impression of the friction ridge pattern. In this work, we propose a new probabilistic framework for Automated Fingermark Quality Assessment (AFQA). We used modern deep learning techniques, which have the ability to extract patterns even from noisy data, and combined them with a methodology from the field of eXplainable AI (XAI) to make our models more transparent. Our solution first predicts a quality probability distribution, from which we then calculate the final quality value and, if needed, the uncertainty of the model. Additionally, we complemented the predicted quality value with a corresponding quality map. We used GradCAM to determine which regions of the fingermark had the largest effect on the overall quality prediction. We show that the resulting quality maps are highly correlated with the density of minutiae points in the input image. Our deep learning approach achieved high regression performance, while significantly improving the interpretability and transparency of the predictions.