Mutations are not completely random as could have been expected from simple probability calculations, and we are off around 10^100. It is not a small problem.
At least they now are finding some bias in the DNA-mutations:

*De novo mutation rates at the single-mutation resolution in a human HBB gene-region associated with adaptation and genetic disease *
paper

While it is known that the mutation rate varies across the genome, previous estimates were based on averaging across various numbers of positions. Here we describe a method to measure the origination rates of target mutations at target base positions and apply it to a 6-bp region in the human hemoglobin subunit beta (HBB) gene and to the identical, paralogous hemoglobin subunit delta (HBD) region in sperm cells from both African and European donors. The HBB region of interest (ROI) includes the site of the hemoglobin S (HbS) mutation, which protects against malaria, is common in Africa and has served as a classic example of adaptation by random mutation and natural selection. We found a significant correspondence between de novo mutation rates and past observations of alleles in carriers, showing that mutation rates vary substantially in a mutation-specific manner that contributes to the site frequency spectrum. We also found that the overall point mutation rate is significantly higher in Africans than in Europeans in the HBB region studied. Finally, the rate of the 20A→T mutation, called the 'HbS mutation' when it appears in HBB, is significantly higher than expected from the genome-wide average for this mutation type. Nine instances were observed in the African HBB ROI, where it is of adaptive significance, representing at least three independent originations; no instances were observed elsewhere. Further studies will be needed to examine mutation rates at the single-mutation resolution across these and other loci and organisms and to uncover the molecular mechanisms responsible.

Mutation bias reflects natural selection in Arabidopsis thaliana
paper

Since the first half of the twentieth century, evolutionary theory has been dominated by the idea that mutations occur randomly with respect to their consequences1. Here we test this assumption with large surveys of de novo mutations in the plant Arabidopsis thaliana. In contrast to expectations, we find that mutations occur less often in functionally constrained regions of the genome—mutation frequency is reduced by half inside gene bodies and by two-thirds in essential genes. With independent genomic mutation datasets, including from the largest Arabidopsis mutation accumulation experiment conducted to date, we demonstrate that epigenomic and physical features explain over 90% of variance in the genome-wide pattern of mutation bias surrounding genes. Observed mutation frequencies around genes in turn accurately predict patterns of genetic polymorphisms in natural Arabidopsis accessions (r = 0.96). That mutation bias is the primary force behind patterns of sequence evolution around genes in natural accessions is supported by analyses of allele frequencies. Finally, we find that genes subject to stronger purifying selection have a lower mutation rate. We conclude that epigenome-associated mutation bias2 reduces the occurrence of deleterious mutations in Arabidopsis, challenging the prevailing paradigm that mutation is a directionless force in evolution.