# Probability of all alleles represented in a sample

I'm trying to wrap my head around some formulas presented in the 1992 paper from Chakraborty Sample Size Requirements for Addressing the Population Genetic Issues of Forensic Use of DNA Typing, but I have not been able to.

Specifically, the right hand side of formula (16) and it's relation with formula (13).

$$1-\sum\limits_{i=1}^{k}(1-p_{i})^{2n}$$ (13)

$$[1-(1-p)^{2n}]^{r}\geqslant1-\alpha$$ (16)

Formula 13 indicates the probability, for a locus with $$k$$ segregating alleles whose frequencies are contained in the vector $$p$$, that all alleles are represented in a given sample of size $$n$$, and the right hand side of formula 16 indicates the probability of $$r$$ alleles to be represented in a given sample of size $$n$$.

First of all, why, based on 13, the expression inside the summation indicates the probability of an allele of frequency p, to remain unobserved in a sample of size n?

I tried to understand this from the Hardy-Weinberg equation but did not have any success.

Second, Why to take the expression in (16) to the r'th power?

Which biological concepts am I missing?

• I don't think there is any biology here, just concepts of probability Sep 28, 2020 at 17:44
• @BryanKrause Probability in this case is been used as a tool for a specific application, like in every other problem involving probability. The main issue here (for me) is to understand the underlying biological concepts to give sense to, for example, why is it that the power in (13) is 2n. Is not a problem related to solve for n in (16) or something of that nature, I'm more interested in understanding the concepts that dictate the reason(s) why the expressions adopt that form to estimate the probability of interest. Sep 28, 2020 at 22:07
• @BryanKrause, I absolutely agree that these are probability questions not pop gen questions per se. If I wanted to answer anyway, is it your opinion that (1) answering here would be harmless or (2) we should tell OP to go away, post it somewhere, leave a link here, and close it? Sep 28, 2020 at 22:16
• @BenBolker Grey area. I think if you want to answer the probability question here I'd appreciate if you narrate it with the biology, and I think that will overall be well received. Sep 29, 2020 at 0:11

I'm going to strictly answer the questions, rather than step through the proof, because it involves a lot of formatting that I'm not familiar with. Other folks are welcome to edit this!

Equation 13

This equation assumes a diploid genotype, given by the $$2n$$ power with $$n$$ individuals. For anything with greater ploidy than mono-, it's mathematically simpler to determine the probability that an allele is not present. As an example, see this calculation of a triploid Hardy-Weinburg equilibrium equation. Using this simplification,

$$P(single$$ $$allele$$ $$not$$ $$present)$$ $$= (1$$ $$- P(allele$$ $$present))$$ ^ $$(ploidy)$$ ^ $$(n)$$

$$= (1$$ $$- P(allele$$ $$present))$$ ^ $$(ploidy$$ * $$n)$$

With $$k$$ segregating alleles, each allele has its own non-presence probability. The probability of total non-presence is $$1 - (sum$$ $$of$$ $$P(each$$ $$non$$-$$presence))$$

Equation 16

In this equation, the author describes the probability that all alleles are present at a given frequency. These allele presences are independent of each other and therefore multiplicative. Since $$P(allele$$ $$present)$$ is vectorized, this product can be simplified to $$^r$$

• Tip - to format equations, you should use the \$ symbol rather than the grave accent. Sep 30, 2020 at 9:09
• Aha, thanks! That's a lot better :) Sep 30, 2020 at 17:03