## Paternity calculation with placental mixtures

Sometimes a paternity test is needed based on a mixture of mother and child.
That is, the embryo died or was aborted, and the only DNA available is the
mother's type, the alleged father's, and a type from the abortus that is
a combination of the mother and the fetus.
In such a situation it's easy enough to work out the proper likelihood
ratios from first principles. Before doing that ...

### Shortcut

It turns out that it comes down to a simple rule that works for calculating any pattern
of the placental mixture situation. Just relate it to a standard paternity pattern
as follows:
- If the mixture has one or two alleles, pretend that the child has that type.
- If the mixture has three alleles, pretend that the child has the non-maternal allele plus either of the others.
- Then compute as a normal paternity calculation.

### Example

Let's check the rule for the pattern
| Mother | PR |

| Mixture | PQR |

| Alleged father | QS |

The paternity index PI=X/Y where

X = Pr(such types | paternity by Alleged father), and

Y = Pr(such types | man unrelated)

as usual. For neatness let M=Pr(mother's type) and F=Pr(Alleged father's type).
Then

X = M•F•Pr(Mixture | parents as alleged) = M•F•1/2;

Y = M•F•Pr(Mixture | Mother & random sperm) = M•F•Pr(Q).

PI = 1/2Pr(Q).

Note that this is the same formula as that for the normal paternity pattern

| Mother | PR |

| Child | PQ |

| Alleged father | QS |

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Ideas in forensic mathematics