# Posers in Forensic Mathematics

The littlest database
Suppose you want to estimate allele frequencies for some DNA locus. How big should the database be? Sometimes N=100 individuals (200 alleles) is suggested as a practical size. But surely N=99 will do almost as well. And if that is so, why not N=98? And so on. Naturally the utility gradually diminishes as N becomes smaller. But for what value of N does the utility disappear completely? What is the absolutely smallest database that is any use at all? And what use is it?

Man, woman, and child
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?

Haplotype frequencies
Suppose you collect a sample of 11-locus Y-chromosome haplotypes from 100 men and every haplotype in the sample is different. We know that the various loci are linked, so it would not be valid to estimate the population frequency of a haplotype by assuming the "multiplication rule" – most haplotypes figure to be more common than the product of the frequencies of the alleles at each locus.

Since multiplication is out, what is a good guess as to the frequency of each haplotype? 1/100?