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Formulas for paternity situations
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Likelihood ratios for sibship and half-sibship

The formulas below were derived with the help of the Symbolic iKnship Program

The triple proportion F:H:U is listed, where F, H, and U are the likelihoods of the genetic types assuming full-shipship*, half-sibship, and unrelated respectively.

From these proportions all likelihood ratios can quickly be written. For example, the likelihood ratio comparing full to half-sibship would be F/H.
Children share two alleles (both are pq) (1+p+q+2pq) : (p+q+4pq) : 8pq
Share two alleles doubly (both are pp) (1+p)2 : 2p(1+p) : (2p)2
Share one allele doubly (they are pp and pq) note (1+p) : (1+2p) : 4p
Share one allele (both heterozygous, pq and pr) note (1+2p) : (1+4p) : 8p
Share no alleles (pq or pp vs. rs or rr) 1 : 2 : 4

Note: Sharing one allele is evidence that the children share at least one parent, but very little evidence that they share one parent rather than two.
typo – as kindly pointed out by Thirsa Kraaijenbrink, Human Genetics, Leiden University Medical Center