Bear in mind that DNA test != full sequencing.

I'm a bit rusty on my combinatorics, but borrowing from wikipedia[1] "if n(p;d) denotes the number of random integers drawn from [1,d] to obtain a probability p that at least two numbers are the same, then:"(...)

We have stated above that n=2, p=1/2(10^(39))=1/(2e27), so we have:

      p ~ 1 - ( (d-1)/d )^(n(n-1)/2))
        = 1 - ( (d-1)/d )^(2(2-1)/2)) # -> 2/2 -> 1
        = 1 - ( (d-1)/d )
     p-1=-(d-1)/d
    (p-1)d=-d+1
     pd-d=1-d  
       pd=1
        d=1/p 
        d~2e27 #Yeah, I guess this should have been obvious, in
                    retrospect

For n=20k=2e4:

     p(n=2e4,d=2e27) ~ 1-((d-1)/d)^(n(n-1)/2)

Which is so small that in Python I had to use the Decimal module to compute p~1e-19 or so.

However, this assumes an even distribution across the possibilities -- and we've approximated the number of combinations based on the statement of collision probability -- not based on actual number of base pairs, or based on some model on how these are likely to be distributed... (eg: if we test for sex, but only sample males -- does that mean we loose a certain number of possibilities? How about caucasian vs other races? It depends on what markers we are testing. Further we know that some markers are likely to match, because there's bound to be some common ansecstors among some members in the sample etc.)

Note that the math above might be entirely bogus -- I'm off to bed, so I'd assume I've made some silly mistakes...

[edit: On actually reading the article, it could appear that the quote on "two billion billions of billions" are indeed meant to indicate he likelihood of someone sharing the same DNA, rather than the likelihood of two samples testing as similar.]

[1] https://en.wikipedia.org/wiki/Birthday_problem#The_generaliz...