Formula Derivation for the Probability Distribution of IBS Score in Unrelated Individual Pairs

To derive the probability equation given by STR allele frequencies of identity by state （IBS） score shared by unrelated individual pairs.

By comparing the STR genotypes of two unrelated individuals, three mutually exclusive combinations could be obtained： （1） sharing 2 identical alleles, a₂=1, otherwise a₂=0; （2） sharing 1 identical allele, a₁=1, otherwise a₁=0; （3） sharing 0 identical allele, a₀=1, otherwise a₀=0. And the IBS score of the one STR locus in this unrelated individual pair could be given by the formula： ibs=2a₂+a₁. The probability of a₂=1 （p₂）, a₁=1 （p₁） and a₀=1 （p₀） were derived and expressed in powers of the allele frequencies. Subsequently, for a genotyping system including n independent STR loci, the characteristics of binomial distribution of IBS score shared by a pair of unrelated individuals could be given by p₂ and p₁ （l=1, 2, …, n）.

All the general equations of p₂, p₁ and p₀ were derived from the basic conceptions of a₂, a₁ and a₀, respectively. Given f （i=1, 2, …, m） as the ith allele frequency of a STR locus, the general equations of p₂, p₁ and p₀ could be respectively expressed in powers of f： [Formula: see text],[Formula: see text] and [Formula: see text]. The sum of p₂, p₁ and p₀ must be equal to 1. Then, the binomial distribution of IBS score shared by unrelated individual pairs genotyped with n independently STR loci could be written by： IBS~B（2n, π）, and the general probability, π, could be given by the formula： [Formula: see text].

In the biological full sibling identification, the probability of null hypothesis corresponding to any specific IBS score can be directly calculated by the general equations presented in this study, which is the basement of the evidence explanation.