If the alleged father is the biological father then this can happen one in four ways, with a probability of 0.25. If the alleged father is not the biological father, then the mother must pass on allele b with probability of 0.5 and the chance that a male other than the alleged father is the father is dependent upon the frequency of allele c (pc) in the population. This gives a likelihood ratio of:
The same process can be used for any of the possible combinations. Consider the version where the alleged father is homozygous (b,b) and the mother heterozygous (a,b) and the child is heterozygous (a,b)
If the alleged father is the biological father then this can happen in two ways, with a probability of 0.5. If the alleged father is not the biological father then the mother must pass on allele b with probability of 0.5 and the chance that a male other than the alleged father is the father is dependent upon the frequency of allele b (pb) in the population. This gives a likelihood ratio of:
Consider a case when the mother is a,b, the child is a,b and the alleged father is a,c.
Allele a or b could be passed from the mother. Note that if she passed on allele a then this would be an exclusion and therefore it would need to be allele b that is passed from mother to child if the man is the biological father. Considering the numerator (Hp) genotype a,b occurs in only one of four ways (0.25). Considering the denominator (Hd) the mother passed on either allele a (0.5) or allele b (0.5) and the chance that either event took place, allele a or allele b, is the sum of the probabilities. This results in the equation below:
In Table 11.1 all the potential combinations of alleles from a mother, child and tested man are shown along with the resulting numerator, denominator and PI equation.
Table 11.1 The numerator and denominator that should be used when calculating a paternity index are determined by the genotypes of the child (G C), mother (G M), and tested man (G TM).
The alleles are represented by i, j, k and 1 where i = j = k = l. Reproduced from Lucy, 2006  p. 174, with permission from John Wiley & Sons (originally based on Evett and Weir, 1998  p. 168)
We can apply the formula in Table 11.1 to the paternity case presented in Table 11.2.
Table 11.2 The result of a paternity test using the Powerplex® 16 STR Kit (Promega). The alleles that the child could have inherited from the mother are underlined and the alleles that are from the biological father are shown in bold. The i,j,k, l symbols correspond to symbols in Table 11.1. The allele frequencies were taken from Marino et al. (2006) 
The combined PI is calculated by applying the product rule and multiplying the PI from each locus in this case the PI is 2920823. This can be represented by this statement:
Statement of positive paternity
The results of the DNA testing are 2920823 times more likely if the tested man is the biological father of the child than if the biological father is another man, unrelated to the tested man.
The significance of likelihood ratios can be difficult for lay people to evaluate and the results are often presented as a probability of paternity, making the results more accessible. To calculate a probability of paternity requires Bayesian analysis and takes into consideration non-genetic evidence: the likelihood ratio (LR) is multiplied by
Table 11.3 The impact of prior probabilities on the probability of pa-ternity is shown with two paternity indexes: one with a value of 1000 and the other taken from the above example, with a value of 2920823
the prior odds of paternity that are determined by non-genetic evidence, such as the testimony of the woman. It can be calculated using equation (11.2).
Taking the above paternity test it is possible to turn the likelihood ratio into a probability of paternity for any prior odds of paternity; for example:
When this figure is used to report the results of a test it is often quoted as a percentage, which is more accessible to non-scientists. In this case the probability of paternity would be quoted as 99.9997%. The value that is attributed to the prior odds of paternity is, of course, subjective. In civil cases, the value of 0.5 is commonly used, although there is little scientific merit to this value. In criminal cases, probabilities of paternity are often not presented because it is the duty of the jury/judge to assess the prior odds of paternity. If results are presented as a probability of paternity, a range of values calculated using different prior odds is often quoted (Table 11.3).