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Major Histocompability Complex (MHC) Class I molecules provide a pathway for cells to present endogenous peptides to the immune system, allowing it to distinguish healthy cells from those infected by pathogens. Software tools based on neural networks such as NetMHC and NetMHCpan predict whether peptides will bind to variants of MHC molecules. These tools are trained with experimental data, consisting of the amino acid sequence of peptides and their observed binding strength. Such tools generally do not explicitly consider hydrophobicity, a significant biochemical factor relevant to peptide binding. It was observed that these tools predict that some highly hydrophobic peptides will be strong binders, which biochemical factors suggest is incorrect. This paper investigates the correlation of the hydrophobicity of 9-mer peptides with their predicted binding strength to the MHC variant HLA-A*0201 for these software tools. Two studies were performed, one using the data that the neural networks were trained on and the other using a sample of the human proteome. A significant bias within NetMHC-4.0 towards predicting highly hydrophobic peptides as strong binders was observed in both studies. This suggests that hydrophobicity should be included in the training data of the neural networks. Retraining the neural networks with such biochemical annotations of hydrophobicity could increase the accuracy of their predictions, increasing their impact in applications such as vaccine design and neoantigen identification.