Cancer research aims to identify genes that cause or control disease progression. Although a wide range of gene sets have been published, they are usually in poor agreement with one another. Furthermore, recent findings from a gene-expression cohort of different cancer types, known as positive random bias, showed that sets of genes chosen randomly are significantly associated with survival time much higher than expected. In this study, we propose a method based on Brouwer’s fixed-point theorem that employs significantly survival-associated random gene sets and reveals a small fixed-point gene set for cancers with a positive random bias property. These sets significantly correspond to cancer-related pathways with biological relevance for the progression and metastasis of the cancer types they represent. Our findings show that our proposed significant gene sets are biologically related to each cancer type available in the cancer genome atlas with the positive random bias property, and by using these sets, positive random bias is significantly more reduced in comparison with state-of-the-art methods in this field. The random bias property is removed in 8 of these 17 cancer types, and the number of random sets of genes associated with survival time is significantly reduced in the remaining 9 cancers.