Man, woman, and child

The problem:

The man may be the father and the woman may be the mother but we are not to assume either one. Some genetic tests are performed and we do the following analyses
1. The woman is compared with the child as a fatherless maternity test, and the likelihood ratio is very high; she looks like a mother to this child.
2. The man, woman, and child are compared as an ordinary paternity trio and the likelihood ratio is very low, perhaps even zero. He does not look like the father if she is the mother.
3. Can we therefore conclude that he is unlikely to be the father regardless of who might be the mother?
How does the reasoning go? Or, is it necessary to make a third, separate analysis – a motherless paternity calculation comparing just the man and the child?

No, the conclusion doesn't necessarily follow. Consider the following simple example:

• Man and woman are both QQ, where Q is rare.
• Child is PQ.

Thus

1. The likelihood ratio for the woman alone is 1/2q, a large number.
2. The likelihood ratio for the trio is 0 – very low.
3. But you cannot predict thereby that the likelihood ratio for the man will be low, because in this case it is 1/2q, not low.