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Skin cancer, primarily resulting from the abnormal growth of skin cells, is among the most common cancer types. In recent decades, the incidence of skin cancer cases worldwide has risen significantly (one in every three newly diagnosed cancer cases is a skin cancer). Such an increase can be attributed to changes in our social and lifestyle habits coupled with devastating man-made alterations to the global ecosystem. Despite such a notable increase, diagnosis of skin cancer is still challenging, which becomes critical as its early detection is crucial for increasing the overall survival rate. This calls for advancements of innovative computer-aided systems to assist medical experts with their decision making. In this context, there has been a recent surge of interest in machine learning (ML), in particular, deep neural networks (DNNs), to provide complementary assistance to expert physicians. While DNNs have a high processing capacity far beyond that of human experts, their outputs are deterministic, i.e., providing estimates without prediction confidence. Therefore, it is of paramount importance to develop DNNs with uncertainty-awareness to provide confidence in their predictions. Monte Carlo dropout (MCD) is vastly used for uncertainty quantification; however, MCD suffers from overconfidence and being miss calibrated. In this paper, we use MCD algorithm to develop an uncertainty-aware DNN that assigns high predictive entropy to erroneous predictions and enable the model to optimize the hyper-parameters during training, which leads to more accurate uncertainty quantification. We use two synthetic (two moons and blobs) and a real dataset (skin cancer) to validate our algorithm. Our experiments on these datasets prove effectiveness of our approach in quantifying reliable uncertainty. Our method achieved 85.65 ± 0.18 prediction accuracy, 83.03 ± 0.25 uncertainty accuracy, and 1.93 ± 0.3 expected calibration error outperforming vanilla MCD and MCD with loss enhanced based on predicted entropy. [ABSTRACT FROM AUTHOR]