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Suppose a specific gene has two possible alleles. In a given population all members have only the dominant one. By a mutation on member receives the first recessive allele of that population. Is it possible that one day the descendants of this population have this allele manifested? I think it's very unlikely that this allele survive until another similar mutation occurs and they get manifested in the phenotype.

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The short answer is: Because of genetic drift. If a mutation does not influence the fitness in heterozygote individuals, then its frequency varies only through genetic drift while it reaches some high enough frequency. If by chance, the frequency of this new allele achieve a high enough level so that the homozygous for this allele given by $x^2$ where $x$ is the frequency of the new allele is high enough to have considerable effect, then the selection coefficient $s$ between the recess if homozygote and the two other genotypes matters.

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