Using the study of Bruce Walsh, I have tried to deduce an estimation of the absolute TMRCA (whatever the surnames are) from the conditional TMRCA (when the surname is shared) computed by FTDNA TiP.

In principle the conditional probability is obtained by dividing the absolute probability by a coefficient equal to the probability for 2 persons to share the same surname, which is equal to the frequency of the surname in a given population. Let F be this frequency.

If we look at the posterior density (formula 12), we can simplify it by a first order approximation, and get a density which only depends on t^(n-k) for n and k fixed.

Now, if we integrate this density, by performing a substitution of the variable t, consisting of multiplying t by a coefficient c, we will dilate the integration interval and multiply the density by (1/c)^(n-k+1).

Consequently, we can get rid of F, if we equal c^(n-k+1) to 1/F, that is by choosing c equal to the (n-k+1) root of 1/F.

Conclusion: TMRCA (absolute) = (1/F)^(1/(n-k+1)) TMRCA (conditional)

I have tried to estimate a value of F by looking at the French population: most of the genealogists consider that there are approximately 350 000 surnames (modulo their variants) for 60 millions of people, corresponding to 171 individuals in a surname cluster. So F=171/60000000, and 1/F= 351000

Turning to my personal case:

At the Y 67 level: one match with n=67 and k=60, so:

c=351000^(1/8)=5

Using FTDNA TiP, I get (conditional TMRCA| absolute TMRCA| percentage)

4 20 23.58%

8 40 68.57%

12 60 91.1%

At the Y 111 level: same match with n=111 and k=110, so:

c=351000^(1/12)=3

4 12 0.65%

8 24 17.71%

12 36 56.61%

16 48 85.5%

It is coherent with my first estimation, based on historical research.